Linear multistep methods and global Richardson extrapolation

TL;DR

This paper explores applying Richardson extrapolation to linear multistep methods for solving ODE initial-value problems, enhancing convergence order and stability without modifying existing codes.

Contribution

It introduces a combined LMM-RE approach that improves convergence order and stability while maintaining compatibility with existing LMM implementations.

Findings

LMM-RE achieves higher order accuracy.

The combined method retains favorable stability properties.

Existing LMM codes can be used without modifications.

Abstract

In this work, we study the application the classical Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for solving initial-value problems of systems of ordinary differential equations numerically. The advantage of the LMM-RE approach is that the combined method possesses higher order and favorable linear stability properties in terms of $A$- or $A(\alpha)$-stability, and existing LMM codes can be used without any modification.