Local Linearization-Runge Kutta Methods: a class of A-stable explicit integrators for dynamical systems

TL;DR

This paper introduces a new class of explicit integrators for ODEs that combine local linearization with Runge-Kutta methods, achieving A-stability and high order accuracy for diverse dynamical systems.

Contribution

It proposes a novel approach to construct high order A-stable explicit integrators by splitting the solution and combining local linearization with existing Runge-Kutta schemes.

Findings

The new schemes are A-stable and high order accurate.

Convergence and dynamical properties are theoretically analyzed.

Performance comparisons show advantages over standard Matlab codes.

Abstract

A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge-Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.