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A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

Zhongcheng Wang
Organization: Shanghai University
Yonghua Ge
Organization: Shanghai University
Yongming Dai
Organization: Shanghai University
Deyin Zhao
Organization: Shanghai University
Journal / Anthology

Computer Physics Communications
Year: 2004
Volume: 160
Page range: 23-45

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods–Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required.

*Applied Mathematics > Numerical Methods

Multi-derivative method, High-order linear two-step methods, Schrödinger equation, Eigenvalue problems, High precision methods, Numerov's method