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ImageSize->{635, 86}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Solving Evolution Equations with ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Title"], Cell["\<\ Rob Knapp Wolfram Research\ \>", "Author"], Cell[CellGroupData[{ Cell["Abstract", "Section"], Cell["\<\ The NDSolve command in Mathematica has a framework for solving \ evolution equations using the numerical method of lines. In this \ presentation, the developer of the framework will explain how it works and \ then show several examples of how to tune the method to get better solutions \ in less time. \ \>", "Text"], Cell[CellGroupData[{ Cell["Initialization", "Subsubsection"], Cell[BoxData[{ \(\(Off[General::spell];\)\), "\[IndentingNewLine]", \(\(Off[General::spell1];\)\)}], "Input", CellLabel->"In[1]:=", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(TalkDirectory\ = \ "\"\)], \ "Input", CellLabel->"In[3]:=", InitializationCell->True], Cell[BoxData[ \("C:\\Web\\RD_HTML\\RKnapp\\Talks\\India"\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[BoxData[ \(\($Path\ = \ Join[$Path, \ Map[ToFileName[ TalkDirectory, \ #] &, \ {"\", \ \ "\"}]];\)\)], "Input", CellLabel->"In[4]:=", InitializationCell->True], Cell[BoxData[ \(Needs["\"]\)], \ "Input", CellLabel->"In[517]:=", InitializationCell->True], Cell[BoxData[ \(Needs["\"]\)], "Input", CellLabel->"In[5]:=", InitializationCell->True], Cell["\<\ monitorwidget = Widget[\"Frame\", { \"title\" -> \"Solution Monitor\", WidgetSpace[5], { WidgetSpace[5], Widget[\"TextField\", {\"text\" -> \"\", \"editable\"->False}, \ Name->\"time\"], WidgetFill[], WidgetSpace[5], Widget[\"Button\", {\"text\" -> \"Stop\", BindEvent[\"action\", Script[ stopRequested = True; ] ]}], WidgetSpace[5]}, Widget[\"ImageLabel\", { \"preferredSize\" -> Widget[\"Dimension\", {\"width\"->288, \ \"height\"->288}] }, WidgetLayout -> {\"Stretching\" -> {Maximize, Maximize}}, Name -> \ \"imagePanel\"], Script[ stopRequested = False; ] }, Name -> \"monitorFrame\"]; \ \>", "Input", CellLabel->"In[6]:=", PageWidth->Infinity, InitializationCell->True, ShowSpecialCharacters->False], Cell["\<\ textmonitorwidget := Widget[\"Frame\", { \"title\" -> \"Solution Monitor\", WidgetSpace[5], { WidgetSpace[5], Widget[\"TextArea\", {\"text\" -> \"\", \"editable\"->True, \ \"rows\"->4, \"columns\"->20}, Name->\"time\"], WidgetFill[], WidgetSpace[5], Widget[\"Button\", {\"text\" -> \"Stop\", BindEvent[\"action\", Script[ stopRequested = True; ] ]}], WidgetSpace[5]}, Script[ stopRequested = False; ] }, Name -> \"monitorFrame\"]; \ \>", "Input", CellLabel->"In[7]:=", PageWidth->Infinity, InitializationCell->True, ShowSpecialCharacters->False] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(The\ Numerical\ Method\ of\ Lines\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ The numerical method of lines (MOL) is a technique for solving \ partial differential equations that works by discretizing in all but one \ dimension, and then integrating the semi-discrete problem as a system of ODEs \ or DAEs. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Advantages\ of\ MOL\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[StyleBox["The method is very general. It can handle\n \ Systems of PDEs\n Systems of PDAEs", "BulletedList"]], "BulletedList"], Cell[TextData[StyleBox["Potential stability difficulties are often handed off \ to ODE methods which have been developed for robustness for a wide class of \ problems.", "BulletedList"]], "BulletedList"], Cell[TextData[StyleBox["The higher order of ODE timestepping methods makes \ the method very flexible.", "BulletedList"]], "BulletedList"], Cell[TextData[StyleBox["Because the time stepping is handled generally, there \ is a lot of flexibility in spatial discretization.", "BulletedList"]], \ "BulletedList"], Cell[TextData[StyleBox["Some adaptivity is achievable with moving mesh \ methods.", "BulletedList"]], "BulletedList"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Disadvantages\ of\ MOL\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[StyleBox["The method only works for PDEs that are well-posed as \ Cauchy (initial value) Problems.\n Elliptic equations do not fit into \ this class.", "BulletedList"]], "BulletedList"], Cell[TextData[StyleBox["Full adaptivity is limited because changing the \ number of grid points requires an ODE method restart.", "BulletedList"]], \ "BulletedList"], Cell[TextData[StyleBox["Some problem geometries are not well suited to the \ method.", "BulletedList"]], "BulletedList"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Burgers`\ equation\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["Burgers` equation,", "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(u\_t = \ \[Nu]\ u\_xx\ - \(\(1\/2\) \(\((u\^2)\)\_x\)\(\ \)\)\)\)\)], "DisplayMath"], Cell["\<\ is a simplified model for some aspects of gas dynamics. When the \ viscosity parameter, \[Nu], is small, the nonlinear wave speed of the \ equation leads to the formation of sharp fronts. It is a good example to \ consider since even though the formation of fronts can lead to numerical \ challenges, there is a transformation (Cole-Hopf) that can be made to find \ analytic solutions such as\ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(u(x, t) = 1\/\(1 + \[ExponentialE]\^\(\(2\ x\ - \ t\)\/\(4\ \[Nu]\)\)\)\)\)\)], \ "DisplayMath"], Cell["\<\ These can be used to check the accuracy of numerical \ approximations.\ \>", "Text"], Cell[TextData[{ "The exact solution ", Cell[BoxData[ \(TraditionalForm\`u(x, t) = 1\/\(1 + \[ExponentialE]\^\(\(2\ x\ - \ t\)\/\(4\ \[Nu]\)\)\)\)]], " is effectively a front which keeps a consistent shape and moves to the \ left with increasing t." }], "Text"], Cell[BoxData[ \(\(u\_exact\ = \ Function[{t, x}, \ \ 1\/\(1\ + \ Exp[\(2\ x\ - \ t\)\/\(4\ \ \[Nu]\)]\)];\)\)], "Input", CellLabel->"In[77]:="], Cell[BoxData[ \(\(Block[{\[Nu]\ = \ 0.01}, Plot3D[u\_exact[t, x], {x, 0, 1}, {t, 0, 2}]];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Burgers`\ equation\ input\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "Here is the equation and boundary conditions as ", StyleBox["Mathematica", FontSlant->"Italic"], " input." }], "Text"], Cell[BoxData[{ \(\(burgers\ = \ D[u[t, x], t]\ \[Equal] \ \ \[Nu]\ D[u[t, x], x, x]\ - \ D[u[t, x]\^2, x]/2\ ;\)\), "\[IndentingNewLine]", \(\(binit\ = \ u[1/2, x]\ \[Equal] \ 1\/\(1\ + \ Exp[\(\(\ \ \)\(2 x\ - \ \(\(1/2\)\(\ \)\)\)\)\/\(4 \ \[Nu]\)]\);\)\), "\[IndentingNewLine]", \(\(bbvals\ = \ {u[t, 0]\ \[Equal] \ 1\/\(1\ + \ Exp[\(-\ \(\(\(\ \)\(t\)\(\ \)\)\/\(4 \ \[Nu]\)\)\)]\), \ u[t, 1]\ \[Equal] \ 1\/\(1\ + \ Exp[\(\(\ \)\(\(2\)\(\ \)\(-\)\(\ \)\(t\)\(\ \ \)\)\)\/\(4 \[Nu]\)]\)};\)\)}], "Input", CellLabel->"In[79]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Burgers`\ equation\ solution\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "The next two commands compute a numerical solution for a particular value \ of \[Nu] and plots the error of the numerical solution as a function of ", StyleBox["x", FontSlant->"Italic"], " for a particular ", StyleBox["t", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[ \(u\_nds\ = \ Block[{\[Nu]\ = \ 0.01}, First[u\ /. \ NDSolve[{burgers, binit, \ bbvals}, u, \ {t, 1/2, 3/2}, \ {x, 0, 1}]]]\)], "Input", CellLabel->"In[82]:="], Cell[BoxData[ \(\(Show[ Block[{\[Nu]\ = \ 0.01, \ t\ = \ 1, \ $DisplayFunction\ = \ Identity}, {Plot[ u\_exact[t, x] - \ u\_nds[t, x], {x, 0, 1}, \ PlotRange \[Rule] All], \ X\ = \ u\_nds@"\"[2]; ListPlot[Transpose[{X, u\_exact[t, X]\ - \ u\_nds[t, X]}], \ PlotStyle \[Rule] RGBColor[1, 0, 0]]}]];\)\)], "Input", CellLabel->"In[83]:="], Cell["\<\ There are two possibly significant sources of the error you see \ above, the spatial discretization error and the temporal discretization \ error. By varying discretization parameters it is possible to see which one \ has the largest contribution. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Static\ spatial\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "With a particular grid, it is relatively easy to create a discretization \ by using replacement rules to replace the continuous function ", Cell[BoxData[ \(TraditionalForm\`u(t, x)\)]], " with a discretized vector function ", Cell[BoxData[ \(TraditionalForm\`U(t)\)]], "." }], "Text"], Cell[TextData[{ "Consider a uniform grid with ", Cell[BoxData[ \(TraditionalForm\`n\ + \ 1\)]], " points and spacing ", Cell[BoxData[ \(TraditionalForm\`h\_n\ = \ 1/n\)]], " for a particular ", Cell[BoxData[ \(TraditionalForm\`n\)]], ". " }], "Text"], Cell[BoxData[ \(n\ = \ 10; h\_n\ = \ 1\/n;\)], "Input", CellLabel->"In[84]:=", CellTags->{"b:7.0", "ndsg:2.0.0"}], Cell["\<\ Let U be the vector function (of t) of the discretized \ values.\ \>", "Text"], Cell[BoxData[ \(\(U[t_]\ = \ Table[u\_i[t], {i, 0, n}];\)\)], "Input", CellLabel->"In[85]:=", CellTags->{"b:7.0", "ndsg:2.0.0"}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Static\ spatial\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Ultimately, the goal is to enable the use of difference formulas of \ any asymptotic order, but at this stage, to keep things simple, consider the \ standard second order difference formulas. \ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(\ u\_x\)(t, \ x)\ \[TildeEqual] \(u(x\ + \ h, \ t)\ - \ \ u(x\ - \ h, \ \ t)\)\/\(2\ h\)\)\)\)], "DisplayMath", CellTags->{"b:7.0", "ndsg:2.0.0"}], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\(u\ \_xx\)(t, \ x)\ \[TildeEqual] \(u(x\ + \ h, \ t)\ - \ 2\ \(u(x, \ t)\)\ - \ \ u(x\ - \ h, \ t)\)\/h\^2\)\)\)], "DisplayMath", CellTags->{"b:7.0", "ndsg:2.0.0"}], Cell[TextData[{ "One easy way to implement such difference formulas in ", StyleBox["Mathematica", FontSlant->"Italic"], " is to use ", StyleBox["ListCorrelate", "MR"], " with the following kernels" }], "Text"], Cell[BoxData[{ \(\(k[1]\ = \ {\(-1\), 0, 1}/\((2\ h\_n)\);\)\), "\[IndentingNewLine]", \(\(k[2]\ = \ {1, \(-2\), \ 1}/h\_n\^2;\)\), "\[IndentingNewLine]", \(\(\(DiscreteD[j_]\)[U_]\ := \ Join[{0}, ListCorrelate[k[j], \ U], {0}];\)\)}], "Input", CellLabel->"In[86]:="], Cell["\<\ Note that this only discretizes the interior. The boundary \ conditions will be handled separately.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Semi\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["The following rules implement the (semi-)discretization", "Text"], Cell[BoxData[ \(\(drules\ = \ {u[t_, x] \[RuleDelayed] \ U[t], \[IndentingNewLine]\(\(Derivative[i_, 0]\)[u]\)[t, x]\ \[RuleDelayed] \ \(\(Derivative[i]\)[U]\)[ t], \[IndentingNewLine]\(\(Derivative[0, j_]\)[u]\)[t, x]\ \[RuleDelayed] \ \(DiscreteD[j]\)[ U[t]], \[IndentingNewLine]x \[Rule] \(h\_n\) Range[0, n]};\)\)], "Input", CellLabel->"In[89]:="], Cell["\<\ ultimately, the goal will be to enable difference formulas of any \ order.\ \>", "Text"], Cell[BoxData[ \(deqns\ = \ Thread[burgers\ /. \ drules]\)], "Input", CellLabel->"In[92]:="], Cell[BoxData[ \(dinit\ = \ Thread[binit\ /. \ drules]\)], "Input", CellLabel->"In[94]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Boundary\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ To handle the boundary conditions, we can replace the right hand \ sides of first and last equations with the time derivatives of the boundary \ conditions\ \>", "Text"], Cell[BoxData[ \(deqns[\([{1, \(-1\)}, \ 2]\)]\ = \ \(D[bbvals, \ t]\)[\([All, 2]\)]\)], "Input", CellLabel->"In[95]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Semi\ discrete\ system\ solution\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ These ODE equations are in a form which can be solved with \ NDSolve\ \>", "Text"], Cell[BoxData[ \(\(lines\ = \ Block[{\[Nu]\ = \ .1}, NDSolve[{deqns, \ dinit}, \ U[t], {t, 1/2, 3/2}]];\)\)], "Input", CellLabel->"In[96]:="], Cell[BoxData[ \(\(lineplot\ = \ Show[\[IndentingNewLine]Block[{$DisplayFunction\ = \ Identity}, \[IndentingNewLine]Table[ ParametricPlot3D[{i\ h\_n, t, \ First[u\_i[t]\ /. lines]}, {t, 1/2, 3/2}], {i, 0, n}]], \ PlotRange \[Rule] All, \ AxesLabel \[Rule] {"\", "\", "\"}];\)\)], "Input", CellLabel->"In[97]:=", CellTags->{"b:7.0", "ndsg:2.0.0"}], Cell["\<\ To compute with a different grid, you can change the definition of \ n above and run through the discretization again. This is not very adaptable \ in that to change the grid you have to work through the entire discretization \ again. Using a symbolic representation for the grid allows going to the next \ level.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Finite\ differences\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "A function which can generate finite difference formulas for any order \ derivative to any degree of approximation for spatial derivatives makes the \ process of discretization easier and more general. Using algorithms for \ generating weights, such as that of Fornberg, it is possible, though not \ simple, to write such a function in ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text"], Cell[TextData[{ "That is not necessary, since in ", StyleBox["Mathematica", FontSlant->"Italic"], " 5, there is a function, ", StyleBox["NDSolve`FiniteDifferenceDerivative", "MR"], " which does this on a grid." }], "Text"], Cell[BoxData[ \(fdd2\ = \ NDSolve`FiniteDifferenceDerivative[ Derivative[2], \ \(h\_n\) Range[0, n]]\)], "Input", CellLabel->"In[98]:="], Cell["\<\ This effectively generates a function which acts on values defined \ on a grid:\ \>", "Text"], Cell[BoxData[ \(TableForm[fdd2[U[t]]]\)], "Input", CellLabel->"In[99]:="], Cell["At the edges of the grid, one-sided derivatives are used.", "Text"], Cell["\<\ The default is to use fourth order differences, however, the order \ can be changed by using an option\ \>", "Text"], Cell[BoxData[ \(TableForm[\(NDSolve`FiniteDifferenceDerivative[ Derivative[2], \(h\_n\) Range[0, n], \ DifferenceOrder \[Rule] 2]\)[ U[t]]]\)], "Input", CellLabel->"In[100]:="], Cell[TextData[{ "More detail can be found in the section on ", ButtonBox["Finite Differences ", ButtonData:>{ FrontEnd`FileName[ {$TopDirectory, "Documentation", "English", "RefGuide", "AdvancedDocumentation", "DifferentialEquations"}, "PDE.nb"], "c:4"}, ButtonStyle->"PageLink"], "in the NDSolve ", ButtonBox["Advanced Documentation", ButtonData:>"Advanced Documentation: NDSolve", ButtonStyle->"RefGuideLinkText"], " in the ", StyleBox["Mathematica", FontSlant->"Italic"], " help browser." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Symbolic\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "A more complete symbolic discretization can be achieved by using a \ function that symbolically represents the finite difference object and is not \ evaluated until the grid and difference order are specified. Let ", Cell[BoxData[ \(TraditionalForm\`X\)]], " represent the grid symbolically and ", Cell[BoxData[ \(TraditionalForm\`U\)]], " represent the discretized function." }], "Text"], Cell[BoxData[ \(\(\(Clear[U, \ X];\)\(\ \)\)\)], "Input", CellLabel->"In[123]:="], Cell["\<\ Then it is simple to define rules which implement the \ discretization\ \>", "Text"], Cell[BoxData[ \(\(rhsrules = \ {\[IndentingNewLine]u[t_, x] \[RuleDelayed] \ U, \[IndentingNewLine]\(\(Derivative[0, j_]\)[u]\)[t, x]\ \[RuleDelayed] \(FiniteDifference[j, \ X, \ m]\)[ U], \[IndentingNewLine]x \[Rule] \ X\[IndentingNewLine]};\)\)], "Input", CellLabel->"In[124]:="], Cell[BoxData[ \(drhs\ = \ burgers[\([2]\)]\ /. \ rhsrules\)], "Input", CellLabel->"In[125]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Symbolic\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ The boundary conditions can be handled much as before. Note the use \ of an intermediate variable to ease the replacement of the appropriate parts.\ \ \>", "Text"], Cell[BoxData[ \(drhsfun\ = \ Function[{t, U}, \ Block[{U1\ = \ drhs1}, U1[\([{1, \(-1\)}]\)]\ = \ dbvals; \ U1]]\ /. \ {drhs1 \[Rule] drhs, \ dbvals\ \[Rule] \ \(D[bbvals, \ t]\)[\([All, 2]\)]}\)], "Input", CellLabel->"In[126]:="], Cell[TextData[{ "This function is not sufficient as is for a proper symbolic discretization \ because, evaluation with symbolic arguments ", Cell[BoxData[ \(TraditionalForm\`t\)]], " and ", Cell[BoxData[ \(TraditionalForm\`U\)]], " will lose necessary information. A wrapper needs to be added which will \ prevent evaluation for symbolic ", Cell[BoxData[ \(TraditionalForm\`t\)]], " and ", Cell[BoxData[ \(TraditionalForm\`U\)]], ". The solution is to use downvalues, for example" }], "Text"], Cell[BoxData[ \(drhsd[t_?NumberQ, \ U_?VectorQ]\ := \ drhsfun[t, U]\)], "Input", CellLabel->"In[127]:="], Cell[TextData[{ "However, the problem with this is that the grid dependent values (in \ particular the finite difference weights) have to be recomputed each time the \ function is evaluated. What is needed is to get the body of the ", StyleBox["Function", "MR"], " above into the downvalues. Doing this is a little bit tricky because of \ the need to prevent evaluation. One handy way to do it is to take advantage \ of the way that patterns substitute in ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Text"], Cell[BoxData[ \(makedrhs[ fun_]\ := \((drhsd[t_?NumberQ, \ U_?VectorQ]\ := fun[t, U])\)\)], "Input", CellLabel->"In[128]:="], Cell[TextData[{ "The desired downvalue is created simply by calling this on the previously \ produced ", StyleBox["Function", "MR"], ":" }], "Text"], Cell[BoxData[ \(makedrhs[drhsfun]\)], "Input", CellLabel->"In[129]:="], Cell[TextData[{ "You can check that the definition is correct by using the ", StyleBox["DownValues", "MR"], " command" }], "Text"], Cell[BoxData[ \(DownValues[drhsd]\)], "Input", CellLabel->"In[130]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Symbolic\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ You can see this is not quite adequate since what we really want to \ do is set up the downvalue once the grid and difference order are defined. \ This, however, can be done with one more layer of rules:\ \>", "Text"], Cell[BoxData[ \(makespecificdrhs[fun_, \ XX_?VectorQ, \ mm_Integer]\ := \ Module[{orders, \ rules}, \ \[IndentingNewLine]orders\ = \ Union[Cases[fun, \ FiniteDifference[j_, __] \[Rule] \ j, Infinity, \ Heads \[Rule] True]]; \[IndentingNewLine]rules\ = \ Map[\ \((FiniteDifference[#, \ X, \ m] -> \ NDSolve`FiniteDifferenceDerivative[Derivative[#], \ XX, \ DifferenceOrder -> mm])\) &, \ orders]; \[IndentingNewLine]makedrhs[\((fun\ /. rules)\)\ /. \ {X \[Rule] XX, \ m \[Rule] mm}]]\)], "Input", CellLabel->"In[143]:="], Cell["Then, for example", "Text"], Cell[BoxData[ \(\(h\_n\) Range[0, n]\)], "Input", CellLabel->"In[136]:="], Cell[BoxData[ \(makespecificdrhs[drhsfun, \ \(h\_n\) Range[0, n], \ 6]\)], "Input", CellLabel->"In[139]:="], Cell[TextData[{ "generates ", StyleBox["DownValues", "MR"] }], "Text"], Cell[BoxData[ \(DownValues[drhsd]\)], "Input", CellLabel->"In[140]:="], Cell[TextData[{ "Producing an efficient evaluation for a vector ", Cell[BoxData[ \(TraditionalForm\`U\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Symbolic\ discretization\ solution\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ All that is needed for the complete symbolic discretization is \ inclusion of the initial conditions, which can readily be included in a \ function which generates the equations to integrate. \ \>", "Text"], Cell[BoxData[ \(sdisc[XX_, \ mm_]\ := \ \((\[IndentingNewLine]makespecificdrhs[drhsfun, \ XX, \ mm]; \[IndentingNewLine]{\(U'\)[t]\ \[Equal] \ drhsd[t, U[t]], \((binit\ /. \ \ u[t_, \ x] \[RuleDelayed] \ \ U[ t])\)\ /. \ x \[Rule] XX})\)\)], "Input", CellLabel->"In[141]:="], Cell["Then, for example", "Text"], Cell[BoxData[ \(sdisc[\(h\_n\) Range[0, n], \ 2]\)], "Input", CellLabel->"In[144]:="], Cell["can be solved by NDSolve", "Text"], Cell[BoxData[ \(sol\ = \ Block[{\[Nu]\ = \ 0.1}, \[IndentingNewLine]NDSolve[ sdisc[\(h\_n\) Range[0, n], \ 2], \ U, \ {t, 1/2, 3/2}]]\)], "Input",\ CellLabel->"In[145]:="], Cell[TextData[{ "This is a vector-valued ", StyleBox["InterpolatingFunction", "MR"], " object. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Symbolic\ discretization\ output\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ One interesting way to visualize it is to plot the data \ points.\ \>", "Text"], Cell[BoxData[ \(Needs["\"]\)], \ "Input", CellLabel->"In[146]:="], Cell[BoxData[{ \(\(ifun\ = \ First[U\ /. \ sol];\)\), "\[IndentingNewLine]", \(\(times\ = \ First[InterpolatingFunctionCoordinates[ ifun]];\)\), "\[IndentingNewLine]", \(\(values\ = \ InterpolatingFunctionValuesOnGrid[ifun];\)\), "\[IndentingNewLine]", \(\(X\ = \ N[\(h\_n\) Range[0, n]];\)\), "\[IndentingNewLine]", \(\(grid\ = \ Outer[List, \ times, \ X];\)\), "\[IndentingNewLine]", \(\(points\ = \ MapThread[Point[Append[Reverse[#1], \ #2]] &, \ {grid, \ values}, \ 2];\)\)}], "Input", CellLabel->"In[147]:="], Cell[BoxData[ \(\(Show[Graphics3D[points]];\)\)], "Input", CellLabel->"In[153]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Generalizing\ the\ symbolic\ discretization\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "There was little done in the above section which was special to Burgers` \ equation. Basically, the only structure which was needed was an equation \ with time derivatives on the left, and initial and Dirichlet boundary \ conditions separated. Periodic boundary conditions are even easier to handle \ because ", StyleBox["NDSolve`FiniteDifferenceDerivative", "MR"], " has a ", StyleBox["PeriodicInterpolation", "MR"], " option which can be set to true for this case. " }], "Text"], Cell[TextData[{ "Neumann and mixed boundary conditions can be handled by using the \ one-sided derivative approximations and ", StyleBox["Solve", "MR"], " with the boundary conditions to find the appropriate differential \ equation to use at the boundary, but are beyond the scope of this paper. The \ general technique is described in the ", ButtonBox["Boundary Conditions ", ButtonData:>{ FrontEnd`FileName[ {$TopDirectory, "Documentation", "English", "RefGuide", "AdvancedDocumentation", "DifferentialEquations"}, "PDE.nb"], "c:6"}, ButtonStyle->"PageLink"], "section in the NDSolve ", ButtonBox["Advanced Documentation", ButtonData:>"Advanced Documentation: NDSolve", ButtonStyle->"RefGuideLinkText"], " in the ", StyleBox["Mathematica", FontSlant->"Italic"], " help browser." }], "Text"], Cell["\<\ Thus, using the symbolic ideas developed above, it is possible to \ write a fairly general discretization function which transforms from a PDE to \ the system of ODEs resulting from discretization. For brevity, the function \ is defined below with only minimal error checking.\ \>", "Text"], Cell[TextData[{ StyleBox["NDSolve", "MR"], " works using the generalized ideas with a few additions. Most \ importantly, ", StyleBox["NDSolve", "MR"], " sets up a symbolic spatial error estimate as a function of the grid and \ determining the grid so that the spatial error on the initial condition \ satisfies the ", StyleBox["AccuracyGoal", "MR"], " and ", StyleBox["PrecisionGoal", "MR"], " tolerances. Description of the technique for obtaining error estimates \ is described in the ", ButtonBox["Spatial Error Estimates ", ButtonData:>{ FrontEnd`FileName[ {$TopDirectory, "Documentation", "English", "RefGuide", "AdvancedDocumentation", "DifferentialEquations"}, "PDE.nb"], "c:7"}, ButtonStyle->"PageLink"], "section in the ", StyleBox["NDSolve", "MR"], " ", ButtonBox["Advanced Documentation", ButtonData:>"Advanced Documentation: NDSolve", ButtonStyle->"RefGuideLinkText"], " in the ", StyleBox["Mathematica", FontSlant->"Italic"], " help browser." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox["Stiffness", "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ The systems of ODEs generated by the method of lines can be quite \ stiff. An example of an equation that leads to stiff ODEs is the KdV equation, \ \ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\(\[Psi]\_\ t + \[Alpha]\ \[Psi]\ \[Psi]\_x + \[Psi]\_xxx = \ 0\)\)\)], "DisplayMath"], Cell["Lets consider it with periodic boundary conditions:", "Text"], Cell[BoxData[{ \(\(KdV\ = \ D[\[Psi][t, x], \ t]\ == \ \(-\[Alpha]\)\ \[Psi][t, x]\ D[\[Psi][t, x], \ x] - D[\[Psi][t, x], \ {x, 3}];\)\), "\[IndentingNewLine]", \(\(KdVinit\ = \ \[Psi][0, x] \[Equal] \ \[CapitalPhi][ x];\)\), "\[IndentingNewLine]", \(\(KdVbc\ = \ \[Psi][t, 0] \[Equal] \ \[Psi][t, 40];\)\)}], "Input", CellLabel->"In[12]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(KdV\ equation\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ An interesting initial condition to consider is a combination of \ solitons\ \>", "Text"], Cell[BoxData[ \(\(S[{\[Alpha]_, \ c0_, \ x0_}, t_, \ x_\ ]\ := \ \(3\ c0\)\/\[Alpha]\ \((Sech[\ \(\@c0\/2\) \((x - x0 - \ c0\ t)\)])\)\^2;\)\)], "Input", CellLabel->"In[15]:="], Cell["\<\ Since we are using periodic boundary conditions add up all the \ period contributions so that the initial condition really does appear \ periodic on the grid. \ \>", "Text"], Cell[BoxData[ \(\(\[CapitalPhi][x_] = S[{\[Alpha], 3, 5}, 0, \ x]\ + S[{\[Alpha], 3/2, 15}, 0, \ x] + \[IndentingNewLine]S[{\[Alpha], 3, 5 + 40}, 0, \ x]\ + S[{\[Alpha], 3/2, 15 + 40}, 0, \ x] + \[IndentingNewLine]S[{\[Alpha], 3, 5 - 40}, 0, \ x]\ + S[{\[Alpha], 3/2, 15 - 40}, 0, \ x];\)\)], "Input", CellLabel->"In[16]:="], Cell[CellGroupData[{ Cell[BoxData[ \(sol\ = \ Block[{\[Alpha]\ = \ 1}, NDSolve[{KdV, \ KdVinit, \ KdVbc}, \ \[Psi], \ {t, 0, \ 10}, {x, 0, 40}]]\)], "Input", CellLabel->"In[17]:="], Cell[BoxData[ RowBox[{\(NDSolve::"mxst"\), \(\(:\)\(\ \)\), "\<\"Maximum number of \ \\!\\(10000\\) steps reached at the point \\!\\(t\\) == \ \\!\\(3.5949462652015947`\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"NDSolve::mxst\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[17]:="], Cell[BoxData[ RowBox[{\(NDSolve::"eerr"\), \(\(:\)\(\ \)\), "\<\"Warning: Scaled local \ spatial error estimate of \\!\\(1821.4531904781345`\\) at \\!\\(t\\) = \ \\!\\(3.5949462652015947`\\) in the direction of independent variable \\!\\(x\ \\) is much greater than prescribed error tolerance. Grid spacing with \ \\!\\(881\\) points may be too large to achieve the desired accuracy or \ precision. A singularity may have formed or you may want to specify a \ smaller grid spacing using the MaxStepSize or MinPoints options. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"NDSolve::eerr\\\"]\\)\"\>"}]], \ "Message", CellLabel->"From In[17]:="], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"\[Psi]", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 3.5949462652015947`}, {"...", 0.`, 40.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output", CellLabel->"Out[17]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(KdV\ stiffness\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["The problems that show up are because of stiffness. ", "Text"], Cell[BoxData[ \(Do[Plot[First[\[Psi][t, x]\ /. \ sol], \ {x, 0, 40}, \ PlotRange \[Rule] {0, .1}], \ {t, 0, 3.5, .5}]\)], "Input", CellLabel->"In[22]:="], Cell[TextData[{ "The default method for NDSolve in ", StyleBox["Mathematica", FontSlant->"Italic"], " switches to a stiff method when stiffness is detected. The default stiff \ methods are BDFs of order 1-5 and are adequate for most problems (they are A-\ \[Alpha] stable). However, for problems with large eigenvalues on the \ imginary axis like this, it is usually necessary to use an A-stable method, \ or one that includes the left hand side of the entire complex plane in its \ stability region." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(KdV\ stiffness\ BDF\ 1, 2\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ BFDs of order 1, 2 are A-stable so one way to remove the stiffness \ problem is to reduce the order to 2\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ sol12\ = \ Block[{\[Alpha]\ = \ 1}, NDSolve[{KdV, \ KdVinit, \ KdVbc}, \ \[Psi], \ {t, 0, \ 10}, {x, 0, 40}, \ Method \[Rule] {"\", \ "\" \ \[Rule] 2}]]]\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(0.7350000000000001`\ Second\), ",", RowBox[{"{", RowBox[{"{", RowBox[{"\[Psi]", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}, {"...", 0.`, 40.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]}], "}"}]], "Output", CellLabel->"Out[18]="] }, Open ]], Cell[BoxData[ \(Plot3D[First[\[Psi][t, x]\ /. \ sol12], \ {x, 0, 40}, {t, 0, 10}, \ Mesh \[Rule] False, \ PlotPoints \[Rule] 200, \ PlotRange \[Rule] All, ViewPoint -> {0.134, \ \(-2.274\), \ 2.502}]\)], "Input", CellLabel->"In[24]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(KdV\ stiffness\ Extrapolation\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Linearly implicit extrapolation methods also provide A-stable \ methods. The StiffnessSwitching method switches to an A-stable method when \ stiffness is detected.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ solss\ = \ Block[{\[Alpha]\ = \ 1}, NDSolve[{KdV, \ KdVinit, \ KdVbc}, \ \[Psi], \ {t, 0, \ 10}, {x, 0, 40}, \ Method \[Rule] "\"]]]\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(4.843999999999999`\ Second\), ",", RowBox[{"{", RowBox[{"{", RowBox[{"\[Psi]", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.`, 10.`}, {"...", 0.`, 40.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]}], "}"}]], "Output", CellLabel->"Out[26]="] }, Open ]], Cell["\<\ For this particular example, it takes longer, but that is because \ the integration of the problem is fairly easy to resolve, so the limitation \ of BDFs to second order is not tragic. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Monitoring\ solutions\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ With this example, the instability showed up right away. For some \ PDEs, it takes much longer to start creeping in. Also, you may just want to check how the solution is evolving for a set of \ initial conditions or parameters and stop it when you are not getting what \ you hoped for.\ \>", "Text"], Cell[TextData[{ "Using GUIKit, it is not difficult to have ", StyleBox["Mathematica", FontSlant->"Italic"], " bring up a window that monitors the solution of the PDE. " }], "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input", CellLabel->"In[52]:="], Cell["The widget defined below sets this up:", "Text"], Cell["\<\ monitorwidget = Widget[\"Frame\", { \"title\" -> \"Solution Monitor\", WidgetSpace[5], { WidgetSpace[5], Widget[\"TextField\", {\"text\" -> \"\", \"editable\"->False}, \ Name->\"time\"], WidgetFill[], WidgetSpace[5], Widget[\"Button\", {\"text\" -> \"Stop\", BindEvent[\"action\", Script[ stopRequested = True; ] ]}], WidgetSpace[5]}, Widget[\"ImageLabel\", { \"preferredSize\" -> Widget[\"Dimension\", {\"width\"->288, \ \"height\"->288}] }, WidgetLayout -> {\"Stretching\" -> {Maximize, Maximize}}, Name -> \ \"imagePanel\"], Script[ stopRequested = False; ] }, Name -> \"monitorFrame\"]; \ \>", "Input", CellLabel->"In[53]:=", PageWidth->Infinity, ShowSpecialCharacters->False] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Monitoring\ solutions\ as\ events\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ In priniciple, the EvaluationMonitor or StepMonitor options of \ NDSolve could be used to give information to the monitor widget, but they are \ less than ideal for this purpose because the times at which the solution \ appears will be unpredictable. \ \>", "Text"], Cell["\<\ A way to have the monitor show regular frames with spacing \ corresponding to the evolution of the temporal variable is to set up an event \ for detection by the EventLocator method.\ \>", "Text"], Cell["\<\ The command below sets up an action to be taken every time the \ event Sin[\[Pi] t/\[CapitalDelta]t] is zero-- i.e. every interval of 0.25 in \ t. The action sends some text, the integration time and the memory used, and an \ image generated by a plot of the solution. (Note it is usually desireable to set a PlotRange since otherwise the image \ will jump around.\ \>", "Text"], Cell[BoxData[{ \(\(\[CapitalDelta]t\ = \ .25;\)\ \), "\[IndentingNewLine]", \(\(event\ = \ Sin[\[Pi]\ t/\[CapitalDelta]t];\)\), "\[IndentingNewLine]", \(action\ := \ \((\[IndentingNewLine]timem\ = \ StringJoin["\", \ ToString[t], \ "\< RAM in use = \>", ToString[ .1 Round[MemoryInUse[]/ 10^5]], \ "\< MB\>"]; \[IndentingNewLine]Sow[{t, \ timem}]; \[IndentingNewLine]monitor\ @\ SetPropertyValue[{"\", \ "\"}, timem]; \[IndentingNewLine]monitor\ @\ InvokeMethod[{"\", "\"}, \ \[IndentingNewLine]ExportString[\ Plot[\[Psi][t, x], \ {x, 0, 40}, \ PlotRange \[Rule] {0, 10}, DisplayFunction \[Rule] \ Identity], \ "\"]]; \[IndentingNewLine]If[\ TrueQ[\ monitor\ @\ Script[stopRequested]], \ monitor\ @\ Script[stopRequested\ = \ False]; Throw[Null, \ "\"]];\[IndentingNewLine])\)\)}], \ "Input", CellLabel->"In[84]:="], Cell[BoxData[ \(monitor\ = \ GUIRun[monitorwidget]; sol\ = \ Block[{\[Alpha]\ = \ 1}, NDSolve[{KdV, \ KdVinit, \ KdVbc}, \ \[Psi], \ {t, 0, \ 10}, {x, 0, 40}, \ Method \[Rule] {"\", \ "\" \[Rule] event, \ "\" \[RuleDelayed] \ action}]]\)], "Input", CellLabel->"In[87]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Spatial\ grid\ points\ and\ DifferenceOrder\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ One of the advantages of MOL is that it makes it relatively easy to \ change the spatial discretization. In this example, I'll show the effect of \ changing the asymptotic order of the spatial derivative approximation.\ \>", \ "Text"], Cell["Here is an equation from Stephen's A New Kind of Science:", "Text"], Cell[BoxData[{ \(\(pde\ = \ D[u[t, x], t, t]\ \[Equal] \ D[u[t, x], x, x]\ + \ \((1\ - \ u[t, x]^2)\) \((1\ + \ 8/3\ u[t, x])\);\)\), "\[IndentingNewLine]", \(\(init\ = \ {u[0, x]\ \[Equal] \ Exp[\(-x^2\)], \ \(\(Derivative[1, 0]\)[u]\)[0, x]\ \[Equal] \ 0};\)\), "\[IndentingNewLine]", \(\ (*\ Periodic\ boundary\ conditions\ *) \ \(bc[ L_]\ := \ {u[t, \ \(-L\)]\ \[Equal] \ u[t, L]};\)\)}], "Input", CellLabel->"In[16]:="], Cell["\<\ A concise way to work with different options is to parameterize the \ whole problem. Here I will parameterize it by the width of the computational domain, L. The goal is to work up to L = 50 or more, but for trial runs, L = 10 is a \ good value to start with.\ \>", "Text"], Cell[BoxData[ \(problem[T_, \ L_]\ := \ Apply[Sequence, \ {{pde, \ init, \ bc[L]}, \ u, \ {t, 0, T}, \ {x, \(-L\), L}}]\)], "Input", CellLabel->"In[19]:="], Cell["For now start with a small trial run:", "Text"], Cell[BoxData[{ \(\(T\ = \ 10;\)\), "\[IndentingNewLine]", \(\(L\ = T;\)\)}], "Input", CellLabel->"In[20]:="], Cell[TextData[{ "This solves it using the default settings of ", StyleBox["Mathematica", FontSlant->"Italic"], ". We have set the step method at this point to make later comparison \ easier.\nWe choose an explicit method since the problem does not lead to any \ particular stiffness. The main reason for this is that the timesteps \ required to resolve solution features are somewhat smaller than those \ required by the stability limit." }], "Text"], Cell[BoxData[ \(\(stepmethod\ = \ "\";\)\)], "Input", CellLabel->"In[22]:="], Cell["This uses the default settings for the method of lines.", "Text"], Cell[BoxData[ \(Timing[ sold\ = \ First[\ u /. \ NDSolve[problem[T, L], \ Method \[Rule] stepmethod]]]\)], "Input",\ CellLabel->"In[23]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Spatial\ grid\ points\ and\ DifferenceOrder\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ The reason the message is issued is because the solution has more \ rapidly varying structure than the intial condition on which the grid is \ based does.\ \>", "Text"], Cell[BoxData[ \(\(Plot3D[sold[t, x], {t, 0, T}, {x, \(-L\), L}, \ PlotPoints \[Rule] 250, \ Mesh \[Rule] False];\)\)], "Input", CellLabel->"In[514]:="], Cell["\<\ When started from zero initial conditions, the equation has a \ peroidically varying solution which you can think of as a background. \ \>", \ "Text"], Cell[BoxData[ \(\(v[t_]\ := \ 3\ Sin[Sqrt[5/6]\ t]^2/\((2\ + \ 3\ Cos[Sqrt[5/6]\ t]^2)\);\)\)], "Input", CellLabel->"In[28]:="], Cell["\<\ Subtracting off the background allows you to see the structural \ changes in the solution more easily.\ \>", "Text"], Cell["\<\ A density plot makes it a bit easier to see some of the interesting \ structure that evolves.\ \>", "Text"], Cell[BoxData[ \(\(DensityPlot[ sold[\(-\[Tau]\), x]\ - \ v[\(-\[Tau]\)], {x, \(-L\), L}, {\[Tau], 0, \(-T\)}, \ PlotPoints \[Rule] 250, \ Mesh \[Rule] False, \ ColorFunction \[Rule] Hue];\)\)], "Input", CellLabel->"In[516]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Spatial\ grid\ points\ and\ DifferenceOrder\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ You can use the information from the message to estimate a better \ starting grid: The default method is asymptotically fourth order. The message reports what the scaled error was. Thus, if nx1 was the points used to compute this, we should try nx2 = \ nx1*(15)^(1/4) \[TildeEqual]2*nx1\ \>", "Text"], Cell["\<\ Load a package of functions for working with the data in \ InterpolatingFunction objects\ \>", "MathCaption"], Cell[BoxData[ \(Needs["\"]\)], \ "Input", CellLabel->"In[36]:="], Cell["This is the length of the spatial grid.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(nx\ = Length[\(InterpolatingFunctionCoordinates[sold]\)[\([2]\)]]\)], "Input",\ CellLabel->"In[518]:="], Cell[BoxData[ \(165\)], "Output", CellLabel->"Out[518]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nx4\ = \ 2*nx\ - \ 1\)], "Input", CellLabel->"In[519]:="], Cell[BoxData[ \(329\)], "Output", CellLabel->"Out[519]="] }, Open ]], Cell["\<\ To keep the commands below more concise, I will set up the options \ for NDSolve that I want to change in a function that allows me to change the \ number of grid points and the difference order. The precision and accuracy goals for the temporal integration as set much \ higher so that the main source of the error in the comparisons shown below is \ from the spatial discretization.\ \>", "Text"], Cell[BoxData[ \(\(opts[n_, \ do_: 4]\ := \ Apply[Sequence, \ \[IndentingNewLine]{PrecisionGoal \[Rule] 8, \ AccuracyGoal \[Rule] 8, \ \[IndentingNewLine]Method -> {"\", \ \[IndentingNewLine]Method \[Rule] stepmethod, \ \ \[IndentingNewLine]"\" \[Rule] \ {"\", \[IndentingNewLine]"\" \[Rule] n, \ "\" \[Rule] n, \[IndentingNewLine]"\" \[Rule] do, \[IndentingNewLine]AccuracyGoal \[Rule] 4, \ PrecisionGoal \[Rule] 4}}}];\)\)], "Input", CellLabel->"In[520]:="], Cell["Solve with a specified number of points.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ sol4\ = \ First[\ u /. \ NDSolve[problem[T, L], \ opts[nx4]]]]\)], "Input", CellLabel->"In[521]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(0.5470000000000255`\ Second\), ",", TagBox[\(InterpolatingFunction[{{0.`, 10.`}, {"...", \(-10.`\), 10.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}]], "Output", CellLabel->"Out[521]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Comparing\ two\ solutions\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ It turns out the the density plots look the same. The difference of \ two computed solutions often gives far more information. It is easier for me to get information from 2 dimensional plots, so I'll be \ comparing solutions at t = T where the error will typically be the largest.\ \ \>", "Text"], Cell[BoxData[ \(\(Plot[sold[T, x]\ - \ sol4[T, x], \ {x, \(-L\), L}, \ PlotRange \[Rule] All];\)\)], "Input", CellLabel->"In[522]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Comparing\ two\ solutions\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ In some situations (not really in this one) , part of what you will \ be seeing in plots like the one above are the effects of interpolating \ between grid points. A way to avoid this is to compare solutions at common \ grid points: this is why the value nx4 = 2*nx1 - 1 was chosen: all points on \ the coarser grid lie on the finer grid.\ \>", "Text"], Cell[BoxData[ \(CompareOnGrid[s1_, \ s2_, \ opts___]\ := \ Module[{tx1\ = InterpolatingFunctionCoordinates[s1], \ tx2\ = \ InterpolatingFunctionCoordinates[s2], diff, \ grid, \ Tend}, \[IndentingNewLine]Tend\ = \ Min[Max[tx1[\([1]\)]], \ Max[tx2[\([1]\)]]]; \[IndentingNewLine]grid\ = \ If[Length[tx1[\([2]\)]]\ > \ Length[tx2[\([2]\)]], \ tx2[\([2]\)], \ tx1[\([2]\)]]; \[IndentingNewLine]diff\ = \ s1[Tend, grid] - \ s2[Tend, \ grid]; \[IndentingNewLine]Show[ Block[{$DisplayFunction = \ Identity}, {\[IndentingNewLine]ListPlot[ Transpose[{grid, \ diff}], \ opts, \ PlotJoined \[Rule] True, PlotRange \[Rule] All], \[IndentingNewLine]ListPlot[ Transpose[{grid, \ diff}], \ opts, \ PlotStyle \[Rule] PointSize[0.015], PlotRange \[Rule] All]\[IndentingNewLine]}]]; \[IndentingNewLine]Max[ Abs[diff]]\[IndentingNewLine]]\)], "Input", CellLabel->"In[37]:="], Cell[CellGroupData[{ Cell[BoxData[ \(CompareOnGrid[sold, \ sol4]\)], "Input", CellLabel->"In[524]:="], Cell[BoxData[ \(0.0012202601171609484`\)], "Output", CellLabel->"Out[524]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Estimating\ the\ error\ by\ extrapolation\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "One way to estimate the error is to use Richardson extrapolation:\nLet \ s(h) be the solution computed with time step h\n ex(h) be the exact \ solution\n p be the method order such that s(h) \[TildeEqual] ex(h) + \ ", Cell[BoxData[ \(TraditionalForm\`ch\^p\)]], "\n Then\n s(h) - s(h/2) \[TildeEqual] ", Cell[BoxData[ \(TraditionalForm\`\(ch\^p\)(1\ - \ 2\^\(-p\))\)]], "\n so\n c \[TildeEqual] ", Cell[BoxData[ FormBox[ StyleBox[\(\(s \((h)\)\ - \ s \((h/2)\)\)\/\(\(h\^p\)( 1\ - \ 2\^\(-p\))\)\), FontSize->18], TraditionalForm]]] }], "Text"], Cell["In the max norm, we have", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c\ = \ 0.0012202601171609484`\/\(\(\((2\ L/nx)\)\^4\) \((15/16)\)\)\)], "Input",\ CellLabel->"In[11]:="], Cell[BoxData[ \(6.029717076681788`\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["So the error on the finer grid is extrapolated to be about ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c\ \((2 L/nx4)\)\^4\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(0.00008234425899229054`\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell["\<\ This can be verified by make a yet finer grid and doing the same \ process.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(nx42\ = \ nx4*2\ - \ 1;\)\), "\[IndentingNewLine]", \(Timing[ sol42\ = \ First[\ u /. \ NDSolve[problem[T, L], \ opts[nx42]]]]\)}], "Input", CellLabel->"In[525]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(2.1090000000000373`\ Second\), ",", TagBox[\(InterpolatingFunction[{{0.`, 10.`}, {"...", \(-10.`\), 10.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}]], "Output", CellLabel->"Out[526]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompareOnGrid[sol4, \ sol42, \ PlotRange \[Rule] All]\)], "Input", CellLabel->"In[527]:="], Cell[BoxData[ \(0.00008471555422384647`\)], "Output", CellLabel->"Out[527]="] }, Open ]], Cell["The extrapolation predicts c = 6.6.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Changing\ the\ difference\ order\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Sometimes a cheaper way of improving the resolution of the solution \ is to increase the difference order instead of making the grid finer.\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ sol6\ = \ First[\ u /. \ NDSolve[problem[T, L], \ opts[nx4, \ 6]]]]\)], "Input",\ CellLabel->"In[529]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(0.6559999999999491`\ Second\), ",", TagBox[\(InterpolatingFunction[{{0.`, 10.`}, {"...", \(-10.`\), 10.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}]], "Output", CellLabel->"Out[529]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompareOnGrid[sol6, \ sol42]\)], "Input", CellLabel->"In[530]:="], Cell[BoxData[ \(4.117573084605475`*^-6\)], "Output", CellLabel->"Out[530]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ sol8\ = \ First[\ u /. \ NDSolve[problem[T, L], \ opts[nx4, \ 8]]]]\)], "Input",\ CellLabel->"In[531]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(1.1879999999999882`\ Second\), ",", TagBox[\(InterpolatingFunction[{{0.`, 10.`}, {"...", \(-10.`\), 10.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}]], "Output", CellLabel->"Out[531]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompareOnGrid[sol8, \ sol42]\)], "Input", CellLabel->"In[532]:="], Cell[BoxData[ \(5.768875503187054`*^-6\)], "Output", CellLabel->"Out[532]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Difference\ order\ limit\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ There are three problems with using arbitrarily high difference \ orders. \ \>", "Text"], Cell["\<\ The coefficents in the finite difference formulas tend to get \ larger with a consequence that the computation of the derivatives involve \ more cancellation, limiting the values h for which the formula are computed \ accurately\ \>", "BulletedList"], Cell["\<\ The expense of computing them goes up with the size of the stencil.\ \ \>", "BulletedList"], Cell["For stiff problems the Jacobian becomes more dense.", "BulletedList"], Cell["\<\ On a periodic grid, however, the limiting difference order can be \ shown to be equivalent to using a trigonometric basis and the derivative \ approximations for this basis can be computed using FFTs which effecitively \ remove the first two problems.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Timing[ solps\ = \ First[\ u /. \ NDSolve[problem[T, L], \ opts[nx4, \ "\"]]]]\)], "Input", CellLabel->"In[533]:="], Cell[BoxData[ RowBox[{"{", RowBox[{\(1.2029999999999745`\ Second\), ",", TagBox[\(InterpolatingFunction[{{0.`, 10.`}, {"...", \(-10.`\), 10.`, "..."}}, "<>"]\), False, Editable->False]}], "}"}]], "Output", CellLabel->"Out[533]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(CompareOnGrid[solps, \ sol8]\)], "Input", CellLabel->"In[534]:="], Cell[BoxData[ \(2.0808370297942247`*^-6\)], "Output", CellLabel->"Out[534]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Bigger\ computation\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["Now set the length of the computational domain to be 50.", "Text"], Cell[BoxData[{ \(\(T\ = \ 50;\)\), "\[IndentingNewLine]", \(\(L\ = \ 50;\)\)}], "Input", CellLabel->"In[24]:="], Cell["\<\ Because of the increasing structure, the initial error estimate \ NDSolve computes will be inadequate. We will compute two solutions and compare them. 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v[\(-\[Tau]\)], {x, \(-L\), L}, {\[Tau], 0, \(-T\)}, \ PlotPoints \[Rule] 250, \ Mesh \[Rule] False, \ ColorFunction \[Rule] Hue];\)\)], "Input", CellLabel->"In[545]:="] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(Model\ for\ turbulent\ \(spectra : \ transforming\ PDEs\)\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ This is a problem I was asked about by someone at the Lawrence \ Livermore National Laboratory in California. He was trying to solve a PDE \ for which the solution gave a model for turbulent spectra. Here is what he \ wrote:\ \>", "Text"], Cell[TextData[{ StyleBox["I am trying to solve a pde for the time dependent energy spectrum \ E(k,t) of an evolving turbulent flow. The pde comes from eqn (3.1) of D.C. \ Besnard et. al., \"Spectral Transport Model for Turbulence,\" Theoretical and \ Computational Fluid Dynamics, Vol 8, pp. 1-35 (1996). It looks like this :\n\ \n", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(\[PartialD]\_t\ E\)\(\ \)\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox["= -", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`c\_1\)], FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_k\ \((k\^\(5/2\)\ \ E\^\(3/2\))\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox["+", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(c\_2\)\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_k\ \((k\^\(7/2\)\ \ \(E\^\(1/ 2\)\) \[PartialD]\_k\ E)\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox["\n\nwhere ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`c\_1\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" and ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`c\_2\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" are constants. The following plot is the initial condition for \ the energy spectrum E(k,0) vs. wavenumber k. It contains energy over a \ narrow band of wavenumbers. Figure 1 of the above reference shows how this \ spectrum evolves over time. I am hoping to start by reproducing this Figure \ using ", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica. 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(What I really want is ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`10\^\(-4\)\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" < k < ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`10\^4\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" )", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[{ \(kmin = 0.01; kmax = 50; npts = 70;\), "\n", \(\(klist = Table[10^logk, {logk, Log[10, kmin], Log[10, kmax], \((Log[10, kmax] - Log[10, kmin])\)/ npts}];\)\)}], "Input", CellLabel->"In[9]:="], Cell[CellGroupData[{ Cell["\<\ c1=1;c2=1; f=NDSolve[{D[y[k, t], t] == -c1*D[(k^2.5)*((y[k, t])^1.5),k]+ c2*D[(k^3.5)*((y[k, t])^0.5)*D[(y[k, t]),k],k], y[k, 0] == z[k], y[kmin, t] == z[kmin], y[kmax, t] == z[kmax]}, y, Join[{k},klist],{t, 0, 100}, MaxSteps->200, Method->StiffnessSwitching]\ \>", "Input", CellLabel->"In[11]:=", CellTags->"S3.9.7"], Cell[BoxData[ RowBox[{\(NDSolve::"eerri"\), \(\(:\)\(\ \)\), "\<\"Warning: Estimated \ initial error on the specified spatial grid in the direction of independent \ variable \\!\\(k\\) exceeds prescribed error tolerance. \\!\\(\\*ButtonBox[\\\ \"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", \ ButtonFrame->None, ButtonData:>\\\"NDSolve::eerri\\\"]\\)\"\>"}]], "Message", CellLabel->"From In[11]:=", CellTags->"S3.9.7"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"y", "\[Rule]", TagBox[\(InterpolatingFunction[{{0.01000000000000001`, 49.99999999999994`}, {0.`, 100.`}}, "<>"]\), False, Editable->False]}], "}"}], "}"}]], "Output", CellLabel->"Out[12]=", CellTags->"S3.9.7"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b\ = \ y\ /. \ First[f]\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ TagBox[\(InterpolatingFunction[{{0.01000000000000001`, 49.99999999999994`}, {0.`, 100.`}}, "<>"]\), False, Editable->False]], "Output", CellLabel->"Out[13]="] }, Open ]], Cell[TextData[{ StyleBox["The following plot shows E(k,t) for t = 0, 0.1, 0.2, 0.7, 10 & \ 100. This looks roughly correct (though not perfect), but when I try to \ increase the range of wavenumber over which I solve this pde, ", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" gets hung up and never returns. 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.96727 .02322 L .96728 .02321 L .96729 .02319 L Mistroke .9673 .02318 L .96731 .02317 L .96732 .02315 L .96733 .02314 L .96734 .02313 L .96735 .02312 L .96736 .0231 L .96737 .02309 L .96738 .02308 L .96739 .02306 L .9674 .02305 L .96741 .02304 L .96742 .02303 L .96743 .02301 L .96744 .023 L .96745 .02299 L .96746 .02297 L .96747 .02296 L .96748 .02295 L .96749 .02294 L .9675 .02292 L .96751 .02291 L .96752 .0229 L .96753 .02288 L .96754 .02287 L .96755 .02286 L .96756 .02285 L .96757 .02283 L .96758 .02282 L .96759 .02281 L .9676 .02279 L .96761 .02278 L .96762 .02277 L .96763 .02276 L .96764 .02274 L .96765 .02273 L .96766 .02272 L .96767 .0227 L .96768 .02269 L .96769 .02268 L .9677 .02266 L .96771 .02265 L .96772 .02264 L .96773 .02263 L .96774 .02261 L .96775 .0226 L .96776 .02259 L .96777 .02257 L .96778 .02256 L .96779 .02255 L Mistroke .9678 .02254 L .96781 .02252 L .96782 .02251 L .96783 .0225 L .96784 .02248 L .96785 .02247 L .96786 .02246 L .96787 .02245 L .96788 .02243 L .96789 .02242 L .9679 .02241 L .96791 .02239 L .96792 .02238 L .96792 .02237 L .96793 .02236 L .96794 .02234 L .96795 .02233 L .96796 .02232 L .96797 .0223 L .96798 .02229 L .96799 .02228 L .968 .02227 L .96801 .02225 L .96802 .02224 L .96803 .02223 L .96804 .02221 L .96805 .0222 L .96806 .02219 L .96807 .02218 L .96808 .02216 L .96809 .02215 L .9681 .02214 L .96811 .02212 L .96812 .02211 L .96813 .0221 L .96814 .02209 L .96815 .02207 L .96816 .02206 L .96817 .02205 L .96818 .02204 L .96819 .02202 L .9682 .02201 L .96821 .022 L .96822 .02198 L .96823 .02197 L .96824 .02196 L .96825 .02194 L .96826 .02193 L .96827 .02192 L .96828 .02191 L Mistroke .96829 .02189 L .9683 .02188 L .96831 .02187 L .96832 .02185 L .96833 .02184 L .96834 .02183 L .96835 .02182 L .96835 .0218 L .96836 .02179 L .96837 .02178 L .96838 .02177 L .96839 .02175 L .9684 .02174 L .96841 .02173 L .96842 .02171 L .96843 .0217 L .96844 .02169 L .96845 .02168 L .96846 .02166 L .96847 .02165 L .96848 .02164 L .96849 .02162 L .9685 .02161 L .96851 .0216 L .96852 .02159 L .96853 .02157 L .96854 .02156 L .96855 .02155 L .96856 .02153 L .96857 .02152 L .96858 .02151 L .96859 .0215 L .9686 .02148 L .96861 .02147 L .96862 .02146 L .96863 .02145 L .96864 .02143 L .96865 .02142 L .96866 .02141 L .96866 .02139 L .96867 .02138 L .96868 .02137 L .96869 .02136 L .9687 .02134 L .96871 .02133 L .96872 .02132 L .96873 .0213 L .96874 .02129 L .96875 .02128 L .96876 .02127 L Mistroke .96877 .02125 L .96878 .02124 L .96879 .02123 L .9688 .02121 L .96881 .0212 L .96882 .02119 L .96883 .02118 L .96884 .02116 L .96885 .02115 L .96886 .02114 L .96887 .02113 L .96888 .02111 L .96889 .0211 L .9689 .02109 L .96891 .02107 L .96891 .02106 L .96892 .02105 L .96893 .02104 L .96894 .02102 L .96895 .02101 L .96896 .021 L .96897 .02098 L .96898 .02097 L .96899 .02096 L .969 .02095 L .96901 .02093 L .96902 .02092 L .96903 .02091 L .96904 .0209 L .96905 .02088 L .96906 .02087 L .96907 .02086 L .96908 .02084 L .96909 .02083 L .9691 .02082 L .96911 .02081 L .96912 .02079 L .96913 .02078 L .96914 .02077 L .96915 .02075 L .96916 .02074 L .96917 .02073 L .96918 .02072 L .96918 .0207 L .96919 .02069 L .9692 .02068 L .96921 .02066 L .96922 .02065 L .96923 .02064 L .96924 .02063 L Mistroke .96925 .02061 L .96926 .0206 L .96927 .02059 L .96928 .02058 L .96929 .02056 L .9693 .02055 L .96931 .02054 L .96932 .02052 L .96933 .02051 L .96934 .0205 L .96935 .02049 L .96936 .02047 L .96937 .02046 L .96938 .02045 L .96939 .02043 L .96939 .02042 L .9694 .02041 L .96941 .0204 L .96942 .02038 L .96943 .02037 L .96944 .02036 L .96945 .02035 L .96946 .02033 L .96947 .02032 L .96948 .02031 L .96949 .0203 L .9695 .02028 L .96951 .02027 L .96952 .02026 L .96953 .02024 L .96954 .02023 L .96955 .02022 L .96956 .02021 L .96957 .02019 L .96958 .02018 L .96959 .02017 L .96959 .02016 L .9696 .02014 L .96961 .02013 L .96962 .02012 L .96963 .0201 L .96964 .02009 L .96965 .02008 L .96966 .02007 L .96967 .02005 L .96968 .02004 L .96969 .02003 L .9697 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.00888 L .97743 .00887 L .97744 .00886 L .97745 .00885 L .97746 .00883 L .97746 .00882 L .97747 .00881 L .97748 .00879 L .97749 .00878 L .9775 .00877 L .97751 .00876 L .97751 .00874 L .97752 .00873 L .97753 .00872 L .97754 .0087 L .97755 .00869 L .97756 .00868 L .97756 .00867 L .97757 .00865 L .97758 .00864 L .97759 .00863 L .9776 .00861 L .9776 .0086 L .97761 .00859 L .97762 .00857 L .97763 .00856 L .97764 .00855 L .97765 .00854 L Mistroke .97765 .00852 L .97766 .00851 L .97767 .0085 L .97768 .00848 L .97769 .00847 L .97769 .00846 L .9777 .00845 L .97771 .00843 L .97772 .00842 L .97773 .00841 L .97774 .00839 L .97774 .00838 L .97775 .00837 L .97776 .00836 L .97777 .00834 L .97778 .00833 L .97778 .00832 L .97779 .0083 L .9778 .00829 L .97781 .00828 L .97782 .00826 L .97783 .00825 L .97783 .00824 L .97784 .00823 L .97785 .00821 L .97786 .0082 L .97787 .00819 L .97787 .00817 L .97788 .00816 L .97789 .00815 L .9779 .00813 L .97791 .00812 L .97792 .00811 L .97792 .0081 L .97793 .00808 L .97794 .00807 L .97795 .00806 L .97796 .00804 L .97796 .00803 L .97797 .00802 L .97798 .00801 L .97799 .00799 L .978 .00798 L .97801 .00797 L .97801 .00795 L .97802 .00794 L .97803 .00793 L .97804 .00791 L .97805 .0079 L .97805 .00789 L Mistroke .97806 .00787 L .97807 .00786 L .97808 .00785 L .97809 .00784 L .97809 .00782 L .9781 .00781 L .97811 .0078 L .97812 .00778 L .97813 .00777 L .97813 .00776 L .97814 .00775 L .97815 .00773 L .97816 .00772 L .97817 .00771 L .97818 .00769 L .97818 .00768 L .97819 .00767 L .9782 .00765 L .97821 .00764 L .97822 .00763 L .97822 .00762 L .97823 .0076 L .97824 .00759 L .97825 .00758 L .97826 .00756 L .97826 .00755 L .97827 .00754 L .97828 .00752 L .97829 .00751 L .9783 .0075 L .97831 .00749 L .97831 .00747 L .97832 .00746 L .97833 .00745 L .97834 .00743 L .97835 .00742 L .97835 .00741 L .97836 .00739 L .97837 .00738 L .97838 .00737 L .97839 .00736 L .97839 .00734 L .9784 .00733 L .97841 .00732 L .97842 .0073 L .97843 .00729 L .97843 .00728 L .97844 .00726 L .97845 .00725 L .97846 .00724 L Mistroke .97847 .00722 L .97848 .00721 L .97848 .0072 L .97849 .00719 L .9785 .00717 L .97851 .00716 L .97852 .00715 L .97852 .00713 L .97853 .00712 L .97854 .00711 L .97855 .00709 L .97856 .00708 L .97856 .00707 L .97857 .00705 L .97858 .00704 L .97859 .00703 L .9786 .00702 L .9786 .007 L .97861 .00699 L .97862 .00698 L .97863 .00696 L .97864 .00695 L .97864 .00694 L .97865 .00692 L .97866 .00691 L .97867 .0069 L .97868 .00689 L .97868 .00687 L .97869 .00686 L .9787 .00685 L .97871 .00683 L .97872 .00682 L .97873 .00681 L .97873 .00679 L .97874 .00678 L .97875 .00677 L .97876 .00675 L .97877 .00674 L .97877 .00673 L .97878 .00671 L .97879 .0067 L .9788 .00669 L .97881 .00668 L .97881 .00666 L .97882 .00665 L .97883 .00664 L .97884 .00662 L .97885 .00661 L .97885 .0066 L .97886 .00659 L Mistroke .97887 .00657 L .97888 .00656 L .97889 .00655 L .97889 .00653 L .9789 .00652 L .97891 .00651 L .97892 .00649 L .97893 .00648 L .97893 .00647 L .97894 .00645 L .97895 .00644 L .97896 .00643 L .97897 .00641 L .97897 .0064 L .97898 .00639 L .97899 .00637 L .979 .00636 L .97901 .00635 L .97901 .00633 L .97902 .00632 L .97903 .00631 L .97904 .0063 L .97905 .00628 L .97905 .00627 L .97906 .00626 L .97907 .00624 L .97908 .00623 L .97909 .00622 L .97909 .0062 L .9791 .00619 L .97911 .00618 L .97912 .00616 L .97913 .00615 L .97913 .00614 L .97914 .00613 L .97915 .00611 L .97916 .0061 L .97917 .00609 L .97917 .00607 L .97918 .00606 L .97919 .00605 L .9792 .00603 L .9792 .00602 L .97921 .00601 L .97922 .00599 L .97923 .00598 L .97924 .00597 L .97924 .00595 L .97925 .00594 L .97926 .00593 L Mistroke .97927 .00592 L .97928 .0059 L .97928 .00589 L .97929 .00588 L .9793 .00586 L .97931 .00585 L .97932 .00584 L .97932 .00582 L .97933 .00581 L .97934 .0058 L .97935 .00578 L .97936 .00577 L .97936 .00576 L .97937 .00574 L .97938 .00573 L .97939 .00572 L .9794 .0057 L .9794 .00569 L .97941 .00568 L .97942 .00567 L .97943 .00565 L .97944 .00564 L .97944 .00563 L .97945 .00561 L .97946 .0056 L .97947 .00559 L .97947 .00557 L .97948 .00556 L .97949 .00555 L .9795 .00553 L .97951 .00552 L .97951 .00551 L .97952 .00549 L .97953 .00548 L .97954 .00547 L .97955 .00545 L .97955 .00544 L .97956 .00543 L .97957 .00541 L .97958 .0054 L .97959 .00539 L .97959 .00537 L .9796 .00536 L .97961 .00535 L .97962 .00533 L .97963 .00532 L .97963 .00531 L .97964 .0053 L .97965 .00528 L .97966 .00527 L Mistroke .97966 .00526 L .97967 .00524 L .97968 .00523 L .97969 .00522 L .9797 .0052 L .9797 .00519 L .97971 .00518 L .97972 .00516 L .97973 .00515 L .97974 .00514 L .97974 .00512 L .97975 .00511 L .97976 .0051 L .97977 .00508 L .97978 .00507 L .97978 .00506 L .97979 .00504 L .9798 .00503 L .97981 .00502 L .97981 .005 L .97982 .00499 L .97983 .00498 L .97984 .00496 L .97985 .00495 L .97985 .00494 L .97986 .00492 L .97987 .00491 L .97988 .0049 L .97989 .00488 L .97989 .00487 L .9799 .00486 L .97991 .00484 L .97992 .00483 L .97993 .00482 L .97993 .0048 L .97994 .00479 L .97995 .00478 L .97996 .00476 L .97996 .00475 L .97997 .00474 L .97998 .00473 L .97999 .00471 L .98 .0047 L .98 .00469 L .98001 .00467 L .98002 .00466 L .98003 .00464 L .98003 .00463 L .98004 .00462 L .98005 .0046 L Mistroke .98006 .00459 L .98007 .00458 L .98007 .00457 L .98008 .00455 L .98009 .00454 L .9801 .00453 L .9801 .00451 L .98011 .0045 L .98012 .00449 L .98013 .00447 L .98014 .00446 L .98014 .00445 L .98015 .00443 L .98016 .00442 L .98017 .00441 L .98018 .00439 L .98018 .00438 L .98019 .00436 L .9802 .00435 L .98021 .00434 L .98022 .00432 L .98022 .00431 L .98023 .0043 L .98024 .00429 L .98025 .00427 L .98025 .00426 L .98026 .00425 L .98027 .00423 L .98028 .00422 L .98028 .00421 L .98029 .00419 L .9803 .00418 L .98031 .00417 L .98032 .00415 L .98032 .00414 L .98033 .00413 L .98034 .00411 L .98035 .0041 L .98036 .00408 L .98036 .00407 L .98037 .00406 L .98038 .00404 L .98039 .00403 L .98039 .00402 L .9804 .00401 L .98041 .00399 L .98042 .00398 L .98043 .00396 L .98043 .00395 L .98044 .00394 L Mistroke .98045 .00393 L .98046 .00391 L .98046 .0039 L .98047 .00388 L .98048 .00387 L .98049 .00386 L .98049 .00384 L .9805 .00383 L .98051 .00382 L .98052 .0038 L .98053 .00379 L .98053 .00378 L .98054 .00376 L .98055 .00375 L .98056 .00374 L .98057 .00372 L .98057 .00371 L .98058 .0037 L .98059 .00368 L .9806 .00367 L .9806 .00366 L .98061 .00364 L .98062 .00363 L .98063 .00362 L .98064 .0036 L .98064 .00359 L .98065 .00358 L .98066 .00356 L .98067 .00355 L .98067 .00354 L .98068 .00352 L .98069 .00351 L .9807 .0035 L .9807 .00348 L .98071 .00347 L .98072 .00345 L .98073 .00344 L .98074 .00343 L .98074 .00341 L .98075 .0034 L .98076 .00339 L .98077 .00337 L .98077 .00336 L .98078 .00335 L .98079 .00333 L .9808 .00332 L .98081 .00331 L .98081 .00329 L .98082 .00328 L .98083 .00327 L Mistroke .98084 .00325 L .98084 .00324 L .98085 .00323 L .98086 .00321 L .98087 .0032 L .98087 .00319 L .98088 .00317 L .98089 .00316 L .9809 .00314 L .98091 .00313 L .98091 .00312 L .98092 .0031 L .98093 .00309 L .98094 .00308 L .98094 .00306 L .98095 .00305 L .98096 .00304 L .98097 .00302 L .98098 .00301 L .98098 .003 L .98099 .00298 L .981 .00297 L .98101 .00296 L .98101 .00294 L .98102 .00293 L .98103 .00291 L .98104 .0029 L .98104 .00289 L .98105 .00287 L .98106 .00286 L .98107 .00285 L .98108 .00283 L .98108 .00282 L .98109 .00281 L .9811 .00279 L .98111 .00278 L .98111 .00277 L .98112 .00275 L .98113 .00274 L .98114 .00273 L .98114 .00271 L .98115 .0027 L .98116 .00269 L .98117 .00267 L .98118 .00266 L .98118 .00264 L .98119 .00263 L .9812 .00262 L .98121 .0026 L .98121 .00259 L Mistroke .98122 .00258 L .98123 .00256 L .98124 .00255 L .98124 .00254 L .98125 .00252 L .98126 .00251 L .98127 .00249 L .98128 .00248 L .98128 .00247 L .98129 .00245 L .9813 .00244 L .98131 .00243 L .98131 .00241 L .98132 .0024 L .98133 .00239 L .98134 .00237 L .98134 .00236 L .98135 .00235 L .98136 .00233 L .98137 .00232 L .98137 .0023 L .98138 .00229 L .98139 .00228 L .9814 .00226 L .98141 .00225 L .98141 .00224 L .98142 .00222 L .98143 .00221 L .98144 .00219 L .98144 .00218 L .98145 .00217 L .98146 .00215 L .98147 .00214 L .98147 .00213 L .98148 .00211 L .98149 .0021 L .9815 .00209 L .9815 .00207 L .98151 .00206 L .98152 .00204 L .98153 .00203 L .98154 .00202 L .98154 .002 L .98155 .00199 L .98156 .00198 L .98157 .00196 L .98157 .00195 L .98158 .00194 L .98159 .00192 L .9816 .00191 L Mistroke .9816 .00189 L .98161 .00188 L .98162 .00187 L .98163 .00185 L .98163 .00184 L .98164 .00183 L .98165 .00181 L .98166 .0018 L .98167 .00178 L .98167 .00177 L .98168 .00176 L .98169 .00174 L .9817 .00173 L .9817 .00172 L .98171 .0017 L .98172 .00169 L .98173 .00167 L .98173 .00166 L .98174 .00165 L .98175 .00163 L .98176 .00162 L .98176 .00161 L .98177 .00159 L .98178 .00158 L .98179 .00157 L .98179 .00155 L .9818 .00154 L .98181 .00152 L .98182 .00151 L .98182 .0015 L .98183 .00148 L .98184 .00147 L .98185 .00145 L .98185 .00144 L .98186 .00143 L .98187 .00141 L .98188 .0014 L .98189 .00139 L .98189 .00137 L .9819 .00136 L .98191 .00135 L .98192 .00133 L .98192 .00132 L .98193 .0013 L .98194 .00129 L .98195 .00128 L .98195 .00126 L .98196 .00125 L .98197 .00123 L .98198 .00122 L Mistroke .98198 .00121 L .98199 .00119 L .982 .00118 L .98201 .00117 L .98201 .00115 L .98202 .00114 L .98203 .00112 L .98204 .00111 L .98204 .0011 L .98205 .00108 L .98206 .00107 L .98207 .00106 L .98207 .00104 L .98208 .00103 L .98209 .00101 L .9821 .001 L .9821 .00099 L .98211 .00097 L .98212 .00096 L .98213 .00094 L .98213 .00093 L .98214 .00092 L .98215 .0009 L .98216 .00089 L .98217 .00087 L .98217 .00086 L .98218 .00085 L .98219 .00083 L .98219 .00082 L .9822 .00081 L .98221 .00079 L .98222 .00078 L .98222 .00076 L .98223 .00075 L .98224 .00074 L .98225 .00072 L .98226 .00071 L .98226 .00069 L .98227 .00068 L .98228 .00067 L .98229 .00065 L .98229 .00064 L .9823 .00063 L .98231 .00061 L .98232 .0006 L .98232 .00058 L .98233 .00057 L .98234 .00056 L .98235 .00054 L .98235 .00053 L Mistroke .98236 .00051 L .98237 .0005 L .98238 .00049 L .98238 .00047 L .98239 .00046 L .9824 .00044 L .98241 .00043 L .98241 .00042 L .98242 .0004 L .98243 .00039 L .98244 .00038 L .98244 .00036 L .98245 .00035 L .98246 .00033 L .98247 .00032 L .98247 .00031 L .98248 .00029 L .98249 .00028 L .9825 .00026 L .9825 .00025 L .98251 .00024 L .98252 .00022 L .98253 .00021 L .98253 .00019 L .98254 .00018 L .98255 .00017 L .98256 .00015 L .98256 .00014 L .98257 .00012 L .98258 .00011 L .98259 .0001 L .98259 8e-005 L .9826 7e-005 L .98261 5e-005 L .98262 4e-005 L .98262 3e-005 L .98263 1e-005 L Mfstroke .98263 1e-005 m .98264 0 L s 0 0 1 r .38634 0 m .39502 .33796 L .40518 .39362 L .413 .41048 L .41934 .41789 L .426 .42221 L .43158 .42395 L .4371 .42447 L .44248 .42406 L .44498 .42361 L .44758 .423 L .45202 .42165 L .45948 .41862 L .47223 .41175 L .48316 .40457 L .49856 .39299 L .53859 .35783 L .56113 .33604 L .57651 .32065 L .58807 .30887 L .59815 .29847 L .60636 .28992 L .61323 .28273 L .6197 .27592 L .62527 .27002 L .63063 .26434 L .63533 .25933 L .6395 .25488 L .64362 .25049 L .6473 .24654 L .65062 .24298 L .65395 .2394 L .65697 .23615 L .66001 .23288 L .66278 .22989 L .66532 .22715 L .6679 .22436 L .67027 .2218 L .67268 .21919 L .67491 .21677 L .67697 .21454 L .67907 .21226 L .68103 .21013 L .68284 .20816 L .68471 .20613 L .68645 .20424 L .68824 .20229 L .68992 .20047 L .69148 .19877 L .69309 .19702 L Mistroke .6946 .19538 L .69615 .19368 L .69761 .19209 L .69898 .19059 L .7004 .18904 L .70173 .18759 L .70299 .18622 L .70429 .1848 L .70551 .18347 L .70678 .18208 L .70798 .18077 L .7091 .17954 L .71028 .17826 L .71139 .17705 L .71243 .1759 L .71352 .17471 L .71455 .17359 L .71562 .17241 L .71663 .1713 L .71759 .17026 L .71859 .16916 L .71953 .16813 L .72052 .16704 L .72146 .16602 L .72234 .16505 L .72327 .16404 L .72414 .16308 L .72497 .16217 L .72584 .16121 L .72666 .16031 L .72753 .15936 L .72834 .15847 L .72912 .15762 L .72993 .15672 L .7307 .15588 L .73144 .15508 L .7322 .15423 L .73293 .15343 L .7337 .15259 L .73442 .15179 L .73511 .15104 L .73584 .15024 L .73653 .14948 L .73725 .14869 L .73794 .14794 L .73859 .14722 L .73927 .14647 L .73993 .14575 L .74055 .14507 L .7412 .14435 L Mistroke .74182 .14367 L .74247 .14295 L .74309 .14227 L .74368 .14162 L .7443 .14094 L .7449 .14029 L .74552 .1396 L .74611 .13895 L .74667 .13833 L .74727 .13768 L .74783 .13706 L .74837 .13646 L .74894 .13584 L .74948 .13524 L .75005 .13462 L .75059 .13402 L .75111 .13345 L .75165 .13285 L .75217 .13228 L .75266 .13174 L .75319 .13117 L .75368 .13062 L .75421 .13004 L .75471 .12949 L .75518 .12897 L .75568 .12841 L .75616 .12789 L .75667 .12733 L .75715 .1268 L .75761 .12629 L .75809 .12576 L .75856 .12525 L .759 .12476 L .75946 .12425 L .75991 .12376 L .76038 .12324 L .76083 .12275 L .76125 .12228 L .7617 .12178 L .76213 .12131 L .76259 .12081 L .76302 .12033 L .76344 .11988 L .76387 .11939 L .76429 .11893 L .76469 .11849 L .76511 .11803 L .76551 .11759 L .76594 .11712 L .76634 .11667 L Mistroke .76673 .11624 L .76714 .11579 L .76753 .11536 L .7679 .11495 L .7683 .11452 L .76868 .1141 L .76907 .11366 L .76945 .11324 L .76982 .11284 L .7702 .11242 L .77057 .11201 L .77096 .11158 L .77133 .11117 L .77168 .11078 L .77206 .11037 L .77242 .10997 L .77276 .10959 L .77312 .10919 L .77347 .10881 L .77384 .10841 L .77419 .10802 L .77452 .10765 L .77487 .10726 L .77521 .10689 L .77554 .10653 L .77588 .10615 L .77621 .10579 L .77655 .10541 L .77688 .10505 L .7772 .1047 L .77754 .10433 L .77786 .10397 L .77819 .1036 L .77852 .10324 L .77883 .1029 L .77916 .10254 L .77947 .10219 L .77977 .10186 L .78009 .10151 L .78039 .10117 L .78072 .10082 L .78102 .10048 L .78132 .10015 L .78163 .09981 L .78193 .09948 L .78224 .09913 L .78255 .0988 L .78283 .09848 L .78314 .09814 L .78343 .09782 L Mistroke .78371 .09751 L .78401 .09718 L .78429 .09687 L .78459 .09653 L .78488 .09622 L .78516 .09591 L .78545 .09559 L .78573 .09528 L .786 .09499 L .78628 .09467 L .78655 .09438 L .78684 .09406 L .78711 .09376 L .78737 .09347 L .78765 .09316 L .78792 .09286 L .7882 .09255 L .78847 .09225 L .78873 .09197 L .789 .09167 L .78927 .09138 L .78952 .0911 L .78978 .0908 L .79004 .09052 L .79031 .09023 L .79057 .08994 L .79081 .08967 L .79108 .08938 L .79133 .0891 L .79157 .08883 L .79182 .08855 L .79207 .08828 L .79233 .088 L .79257 .08772 L .79281 .08746 L .79306 .08718 L .7933 .08692 L .79356 .08664 L .7938 .08637 L .79404 .08611 L .79428 .08583 L .79452 .08557 L .79475 .08532 L .79499 .08505 L .79522 .0848 L .79547 .08453 L .7957 .08427 L .79593 .08402 L .79616 .08376 L .79639 .0835 L Mistroke .79663 .08324 L .79686 .08298 L .79709 .08273 L .79732 .08247 L .79755 .08223 L .79776 .08199 L .79799 .08173 L .79821 .08149 L .79844 .08123 L .79867 .08099 L .79888 .08075 L .79911 .0805 L .79932 .08026 L .79953 .08003 L .79975 .07979 L .79996 .07956 L .80019 .07931 L .8004 .07907 L .80061 .07884 L .80083 .0786 L .80104 .07837 L .80126 .07813 L .80147 .07789 L .80167 .07767 L .80189 .07743 L .8021 .0772 L .80229 .07698 L .8025 .07675 L .80271 .07652 L .80292 .07629 L .80312 .07606 L .80332 .07584 L .80353 .07561 L .80373 .07539 L .80394 .07516 L .80414 .07494 L .80434 .07472 L .80454 .07449 L .80474 .07427 L .80493 .07406 L .80513 .07384 L .80532 .07363 L .80553 .0734 L .80572 .07319 L .80591 .07298 L .80611 .07276 L .8063 .07255 L .80649 .07235 L .80668 .07213 L .80687 .07192 L Mistroke .80706 .07171 L .80725 .0715 L .80743 .0713 L .80763 .07108 L .80781 .07088 L .80801 .07066 L .80819 .07045 L .80838 .07026 L .80857 .07004 L .80875 .06984 L .80892 .06965 L .80911 .06944 L .80929 .06924 L .80948 .06903 L .80966 .06883 L .80984 .06864 L .81002 .06844 L .8102 .06824 L .81037 .06805 L .81055 .06785 L .81072 .06766 L .81091 .06746 L .81108 .06726 L .81125 .06708 L .81143 .06688 L .8116 .06669 L .81178 .06649 L .81196 .06629 L .81213 .06611 L .8123 .06591 L .81247 .06572 L .81264 .06554 L .81281 .06535 L .81298 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"SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ The discretization works essentially the same in multiple spatial \ dimensions as it does in one.\ \>", "Text"], Cell["However, there are a few things to keep in mind:", "Text"], Cell["The discretization leads to much bigger sets of ODEs", "BulletedList"], Cell["\<\ The Jacobian matrices are still very sparse, but are not banded as \ in the 1-D case. Especially for 3 spatial dimensions, iterative methods may \ do much better than direct methods because of fill-in. However, iterative \ methods often need preconditioning, which is a bit of an art.\ \>", \ "BulletedList"], Cell["\<\ Boundary conditions need to be specified normal to the \ computational domain.\ \>", "BulletedList"], Cell["\<\ For now, the spatial computational domain must rectangular, though \ with mappings, different geometries are possible.\ \>", "BulletedList"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageFirst"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPagePrevious"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageNext"]}], ButtonStyle->"SlideHyperlink", ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>FrontEndExecute[ { FrontEndToken[ FrontEnd`SelectedNotebook[ ], "ScrollPageLast"]}], ButtonStyle->"SlideHyperlink"], StyleBox[\(2\ + \ 1\ dimensional\ example\), "Text", FontSize->16, FontWeight->"Bold"], ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], \(\(\ \)\(of\)\(\ \)\), CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], ButtonFrame->"None"]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "This example comes from Michael Trott, who has been instrumental in \ pushing the PDE solving capabilities of ", StyleBox["Mathematica", FontSlant->"Italic"], " to the limit." }], "Text"], Cell[BoxData[ \(<< PSDirect`\)], "Input", CellLabel->"(Dev-NumericalFunction) In[8]:="], Cell[BoxData[ RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "\[CapitalDelta]TC", ",", "\[Xi]", ",", "\[Eta]", ",", "\[ScriptF]", ",", "SEqs", ",", "packet", ",", "nds", ",", "ur", ",", "ui", ",", "makeColoredPolygon", ",", "makeAnimationFrame", ",", "\[IndentingNewLine]", StyleBox[\( (*system\ parameters*) \), FontColor->RGBColor[0, 0, 1]], \(pp\[Xi] = 320\), ",", \(pp\[Eta] = 40\), ",", \(\[CapitalDelta]T = 1/8\), ",", \(L = 5\), ",", \(TMax = 7\)}], "}"}], ",", "\[IndentingNewLine]", StyleBox[\( (*Laplace\ operator\ in\ new\ coordinate\ system*) \), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", RowBox[{\(\(\[CapitalDelta]TC[f_]\)[ u_, {\[Xi]_, \[Eta]_}] := \((D[ u, \[Eta], \[Eta]] - 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Cell[StyleData["Print"], CellMargins->{{66, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, AutoItalicWords->{}, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Print", "SlideShow"], CellMargins->{{100, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontSize->18], Cell[StyleData["Print", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{4, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, LanguageCategory->None, FormatType->InputForm, CounterIncrements->"Graphics", ImageMargins->{{43, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->10], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "SlideShow"], ImageMargins->{{100, Inherited}, {Inherited, 0}}], Cell[StyleData["Graphics", "Printout"], ImageMargins->{{30, Inherited}, {Inherited, 0}}, Magnification->0.8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontColor->RGBColor[0.23621, 0.628321, 0.559747]], Cell[StyleData["CellLabel", "Presentation"], FontSize->12], Cell[StyleData["CellLabel", "SlideShow"], FontSize->12], Cell[StyleData["CellLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["FrameLabel"], LanguageCategory->None, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9], Cell[StyleData["FrameLabel", "Presentation"], FontSize->12], Cell[StyleData["FrameLabel", "SlideShow"], FontSize->12], Cell[StyleData["FrameLabel", "Printout"], FontFamily->"Courier", FontSize->8, FontSlant->"Italic", FontColor->GrayLevel[0]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Presentation Specific", "Section"], Cell[CellGroupData[{ Cell[StyleData["BulletedList"], CellMargins->{{45, 10}, {7, 7}}, CellFrameLabels->{{ Cell[ "\[FilledSmallSquare]", "BulletedList", CellBaseline -> Baseline], Inherited}, {Inherited, Inherited}}, InputAutoReplacements->{"TeX"->StyleBox[ RowBox[ {"T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "LaTeX"->StyleBox[ RowBox[ {"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-.35999999999999999, \ -.10000000000000001}, {0, 0}}, BoxBaselineShift -> -.20000000000000001], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-.074999999999999997, \ -.085000000000000006}, {0, 0}}, BoxBaselineShift -> .5], "X"}]], "mma"->"Mathematica", "Mma"->"Mathematica", "MMA"->"Mathematica", "gridMathematica"->FormBox[ RowBox[ {"grid", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], "webMathematica"->FormBox[ RowBox[ {"web", AdjustmentBox[ StyleBox[ "Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-.17499999999999999, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements->"BulletedList", FontFamily->"Helvetica"], Cell[StyleData["BulletedList", "Presentation"], CellMargins->{{56, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->18], Cell[StyleData["BulletedList", "SlideShow"], CellMargins->{{85, 50}, {10, 10}}, FontSize->18], Cell[StyleData["BulletedList", "Printout"], CellMargins->{{2, 2}, {6, 6}}, TextJustification->0.5, Hyphenation->True, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{139, 27}, {2, 20}}, FontFamily->"Times", FontSize->24, FontSlant->"Italic"], Cell[StyleData["Author", "Presentation"], CellMargins->{{198, 27}, {2, 25}}, FontSize->32], Cell[StyleData["Author", "SlideShow"], CellMargins->{{198, 27}, {2, 50}}, FontSize->32], Cell[StyleData["Author", "Printout"], CellMargins->{{100, 27}, {2, 20}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Affiliation"], CellMargins->{{141, 27}, {30, 12}}, FontFamily->"Times", FontSize->24, FontSlant->"Italic"], Cell[StyleData["Affiliation", "Presentation"], CellMargins->{{198, 27}, {35, 10}}, FontSize->32], Cell[StyleData["Affiliation", "SlideShow"], CellMargins->{{198, 27}, {100, 10}}, FontSize->32], Cell[StyleData["Affiliation", "Printout"], CellMargins->{{100, 27}, {2, 12}}, FontSize->14] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Header Graphic", "Section"], Cell[CellGroupData[{ Cell[StyleData["ConferenceGraphicCell"], ShowCellBracket->True, CellMargins->{{0, 0}, {0, 0}}, Evaluatable->False, PageBreakBelow->False, CellFrameMargins->False, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, Background->RGBColor[0.827451, 0.333333, 0.258824], Magnification->1], Cell[StyleData["ConferenceGraphicCell", "Presentation"], ShowCellBracket->False], Cell[StyleData["ConferenceGraphicCell", "SlideShow"], ShowCellBracket->False], Cell[StyleData["ConferenceGraphicCell", "Printout"], FontSize->8, Magnification->0.75] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GraphicNoMagnification"], CellMargins->{{60, 10}, {7, 7}}, LineSpacing->{1, 3}, CounterIncrements->"Text", FontFamily->"Helvetica", Magnification->1], Cell[StyleData["GraphicNoMagnification", "Presentation"], CellMargins->{{72, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->17], Cell[StyleData["GraphicNoMagnification", "SlideShow"], CellMargins->{{100, 50}, {10, 10}}, FontSize->17], Cell[StyleData["GraphicNoMagnification", "Printout"], CellMargins->{{2, 2}, {6, 6}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Inline Formatting", "Section"], Cell["\<\ These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.\ \>", "Text"], Cell[StyleData["RM"], StyleMenuListing->None, FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["BF"], StyleMenuListing->None, FontWeight->"Bold"], Cell[StyleData["IT"], StyleMenuListing->None, FontSlant->"Italic"], Cell[StyleData["TR"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["TI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["TB"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["TBI"], StyleMenuListing->None, FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement"->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, StyleMenuListing->None, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Italic"], Cell[StyleData["SR"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], Cell[StyleData["SO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SB"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], Cell[StyleData["SBO"], StyleMenuListing->None, FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic"], Cell[CellGroupData[{ Cell[StyleData["SO10"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->10, FontWeight->"Plain", FontSlant->"Italic"], Cell[StyleData["SO10", "Printout"], StyleMenuListing->None, FontFamily->"Helvetica", FontSize->7, FontWeight->"Plain", FontSlant->"Italic"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["InlineFormula"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->1, SingleLetterItalics->True, StyleMenuListing->None], Cell[StyleData["InlineFormula", "Presentation"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "SlideShow"], LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{60, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, HyphenationOptions->{"HyphenationCharacter"->"\[Continuation]"}, LanguageCategory->"Formula", ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "SlideShow"], CellMargins->{{100, 50}, {Inherited, Inherited}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellFrame->{{0, 0}, {0.5, 0.5}}, CellMargins->{{60, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", ScriptLevel->1, FontFamily->"Courier"], Cell[StyleData["Program", "Presentation"], CellMargins->{{72, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "SlideShow"], CellMargins->{{100, 50}, {10, 10}}, LineSpacing->{1, 5}, FontSize->16], Cell[StyleData["Program", "Printout"], CellMargins->{{2, 0}, {6, 6}}, FontSize->9] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0.376471, 0.490196], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonFrame->"None", ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"], FontSize->16], Cell[StyleData["Hyperlink", "SlideShow"]], Cell[StyleData["Hyperlink", "Printout"], FontSize->10, FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"], FontSize->16], Cell[StyleData["MainBookLink", "SlideShow"]], Cell[StyleData["MainBookLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"], FontSize->16], Cell[StyleData["AddOnsLink", "SlideShow"]], Cell[StyleData["AddOnsLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLink", "SlideShow"]], Cell[StyleData["RefGuideLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLinkText"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuide", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLinkText", "Presentation"], FontSize->16], Cell[StyleData["RefGuideLinkText", "SlideShow"]], Cell[StyleData["RefGuideLinkText", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"], FontSize->16], Cell[StyleData["GettingStartedLink", "SlideShow"]], Cell[StyleData["GettingStartedLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DemosLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Demos", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["DemosLink", "SlideShow"]], Cell[StyleData["DemosLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["TourLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "Tour", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["TourLink", "SlideShow"]], Cell[StyleData["TourLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"], FontSize->16], Cell[StyleData["OtherInformationLink", "SlideShow"]], Cell[StyleData["OtherInformationLink", "Printout"], FontSize->10, FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["MasterIndexLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0.269993, 0.308507, 0.6], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MasterIndex", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MasterIndexLink", "SlideShow"]], Cell[StyleData["MasterIndexLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, DefaultNewInlineCellStyle->"None", LanguageCategory->"NaturalLanguage", StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define standard \ ButtonFunctions, for use in palette buttons.\ \>", "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, Placeholder]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["EvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[ ], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[ ], All]}]&)}], Cell[StyleData["CopyEvaluateCell"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[ ], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[ ], All]}]&)}] }, Closed]], Cell[CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Placeholder->True, StyleMenuListing->None, FontSlant->"Italic", FontColor->RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "SlideShow"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["PrimaryPlaceholder"], StyleMenuListing->None, DrawHighlighted->True, FontSlant->"Italic", Background->RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions->{Editable->False, Selectable->False, StripWrapperBoxes->False}], Cell[StyleData["PrimaryPlaceholder", "Presentation"]], Cell[StyleData["PrimaryPlaceholder", "SlideShow"]], Cell[StyleData["PrimaryPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, Hyphenation->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], InputAutoReplacements->{}, AllowInlineCells->False, Hyphenation->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], InputAutoReplacements->{ "->"->"\[Rule]", ":>"->"\[RuleDelayed]", "<="->"\[LessEqual]", ">="->"\[GreaterEqual]", "!="->"\[NotEqual]", "=="->"\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement"->True, LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], LanguageCategory->"Formula", ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[0.964706, 0.929412, 0.839216]] }, Closed]], Cell[CellGroupData[{ Cell["Automatic Styles", "Section"], Cell["\<\ The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. 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