On the stability of Approximate Taylor methods for ODE and their relationship with Runge-Kutta schemes

TL;DR

This paper investigates the stability of Approximate Taylor methods for solving ODEs, confirming their stability regions match those of exact Taylor methods and establishing their relationship with Runge-Kutta schemes.

Contribution

It proves that the stability regions of Approximate Taylor methods are identical to those of their exact counterparts and clarifies their connection to Runge-Kutta methods.

Findings

Stability regions of Approximate Taylor methods match those of exact Taylor methods.

Confirmed the relationship between Approximate Taylor and Runge-Kutta schemes.

Validated conjectures from previous studies about these methods.

Abstract

In [Baeza et al., Computers and Fluids, 159, 156--166 (2017)] a new method for the numerical solution of ODEs is presented. This methods can be regarded as an approximate formulation of the Taylor methods and it follows an approach that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their high order derivatives, are needed. In this reference, the absolute stability region of the new methods is conjectured to be coincident with that of their exact counterparts. There is also a conjecture about their relationship with Runge-Kutta methods. In this work we answer positively both conjectures.