A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods

Abstract

This paper revisits the Taylor method for the numerical integration of initial value problems of Ordinary Differential Equations (ODEs). The main goal is to present a computer program that outputs a specific numerical integrator for a given set of ODEs. The generated code includes a function to compute the jet of derivatives of the solution up to a given order plus adaptive selection of order and step size at run time. The package provides support for several extended precision arithmetics, including user-defined types.

The paper discusses the performance of the resulting integrator in some examples, showing that it is very competitive in many situations. This is especially true for integrations that require extended precision arithmetic. The main drawback is that the Taylor method is an explicit method, so it has all the limitations of these kind of schemes. For instance, it is not suitable for stiff systems.

Keywords:

• 65L05
• Taylor method
• automatic differentiation
• software