Reduced temporal convergence rates in high-order splitting schemes

TL;DR

This paper investigates how high-order splitting schemes with complex coefficients perform less efficiently in certain parabolic equations, especially when diffusion is less stiff than reaction terms, providing explanations for this behavior.

Contribution

It identifies the reduced convergence rates of high-order splitting schemes with complex coefficients in specific parabolic problems and offers a simple explanation for this phenomenon.

Findings

Reduced convergence rates observed in certain parabolic equations.

High-order splitting schemes are most efficient when diffusion is less stiff.

Provides a simple explanation for the reduced convergence behavior.

Abstract

Recently-derived high-order splitting schemes with complex coefficients are shown to exhibit reduced convergence rates for certain parabolic evolution equations. When applied to semilinear reaction-diffusion equations with periodic boundary conditions, these splitting schemes are most efficient when the diffusion terms are much less stiff than the reaction terms. An explanation for this is given in a simple setting.