Linear multistep methods with repeated global Richardson extrapolation

TL;DR

This paper explores the use of repeated global Richardson extrapolation to enhance the convergence order of linear multistep methods for solving initial-value problems, enabling parallel implementation and preserving stability.

Contribution

It introduces a new accelerated sequence with higher convergence order using repeated global Richardson extrapolation applied to linear multistep methods.

Findings

Accelerated sequence achieves order p+l when RE is applied l times.

LMM-RGRE methods can be implemented in parallel without code modifications.

Stability properties of the base methods are preserved in the accelerated methods.

Abstract

In this work, we further investigate the application of the well-known Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for numerically solving initial-value problems of systems of ordinary differential equations. By extending the ideas of our previous paper, we now utilize some advanced versions of RE in the form of repeated RE (RRE). Assume that the underlying LMM -- the base method -- has order $p$ and RE is applied $l$ times. Then we prove that the accelerated sequence has convergence order $p+l$. The version we present here is global RE (GRE, also known as passive RE), since the terms of the linear combinations are calculated independently. Thus, the resulting higher-order LMM-RGRE methods can be implemented in a parallel fashion and existing LMM codes can directly be used without any modification. We also investigate how the linear stability properties of the base method (e.g. $A$- or $A(\alpha)$-stability) are preserved by the LMM-RGRE methods.