Approximate Taylor methods for ODEs

TL;DR

This paper introduces approximate Taylor methods for solving ODEs that are easier to implement than classical Taylor methods, requiring only function evaluations, and can achieve high order with potentially lower computational costs.

Contribution

The paper proposes an approximate Taylor method for ODEs that simplifies implementation and enables high-order schemes without needing derivatives, contrasting with traditional Taylor and Runge-Kutta methods.

Findings

Achieves comparable accuracy to exact Taylor methods

Easier implementation than classical Taylor methods

Potentially lower computational cost in many cases

Abstract

A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge-Kutta schemes. Compared to Runge-Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansion based on exact derivatives. The numerical results that are obtained with our proposal are satisfactory and show that this approximate approach can attain results as good as the exact Taylor procedure with less implementation and computational effort.