Analysis of the effect of Time Filters on the implicit method: increased accuracy and improved stability

TL;DR

This paper explores the use of time filters in linear multistep methods, enhancing accuracy and stability with minimal code changes, and develops a second-order, strongly A-stable method based on backward Euler.

Contribution

It introduces a modular time filtering approach for linear multistep methods, specifically developing a second-order, strongly A-stable method from backward Euler.

Findings

Enhanced stability and accuracy with time filters

Simple implementation requiring minimal code changes

Development of a second-order, strongly A-stable method

Abstract

This report considers linear multistep methods through time filtering. The approach has several advantages. It is modular and requires the addition of only one line of additional code. Error estimation and variable timesteps is straightforward and the individual effect of each step\ is conceptually clear. We present its development for the backward Euler method and a curvature reducing time filter leading to a 2-step, strongly A-stable, second order linear multistep method.