Mathematica 9 is now available

Wolfram Library Archive


Courseware Demos MathSource Technical Notes
All Collections Articles Books Conference Proceedings
Title

Algorithm 924: TIDES, a Taylor Series Integrator for Differential EquationS
Authors

ALBERTO ABAD
ROBERTO BARRIO
FERNANDO BLESA
MARCOS RODRIGUEZ
Journal / Anthology

ACM Transactions on Mathematical Software
Year: 2012
Volume: 39
Issue: 1
Description

This article introduces the software package TIDES and revisits the use of the Taylor series method for the numerical integration of ODEs. The package TIDES provides an easy-to-use interface for standard double precision integrations, but also for quadruple precision and multiple precision integrations. The motivation for the development of this package is that more and more scientific disciplines need very high precision solution of ODEs, and a standard ODE method is not able to reach these precision levels. The TIDES package combines a preprocessor step in Mathematica that generates Fortran or C programs with a library in C. Another capability of TIDES is the direct solution of sensitivities of the solution of ODE systems, which means that we can compute the solution of variational equations up to any order without formulating them explicitly. Different options of the software are discussed, and finally it is compared with other well-known available methods, as well as with different options of TIDES. From the numerical tests, TIDES is competitive, both in speed and accuracy, with standard methods, but it also provides new capabilities.
Subjects

*Mathematics
*Mathematics > Calculus and Analysis > Differential Equations
Keywords

Ordinary Differential Equations, General-Multiple precision arithmetic, Quadrature and Numerical Differentiation-Automatic differentiation, Algorithms, Taylor series method, automatic differentiation, high precision, variational equations, numerical integration of ODEs