A Convergence Analysis of the Peaceman--Rachford Scheme for Semilinear Evolution Equations

TL;DR

This paper provides a convergence analysis of the Peaceman--Rachford scheme for semilinear evolution equations, including cases without Lipschitz nonlinearities, with theoretical results supported by numerical experiments.

Contribution

It extends convergence analysis of the Peaceman--Rachford scheme to dissipative equations without assuming Lipschitz continuity of nonlinearities.

Findings

First and second order convergence depending on solution regularity

Convergence results applicable to Lie scheme in Banach spaces

Numerical experiments confirm theoretical convergence rates

Abstract

The Peaceman--Rachford scheme is a commonly used splitting method for discretizing semilinear evolution equations, where the vector fields are given by the sum of one linear and one nonlinear dissipative operator. Typical examples of such equations are reaction-diffusion systems and the damped wave equation. In this paper we conduct a convergence analysis for the Peaceman--Rachford scheme in the setting of dissipative evolution equations on Hilbert spaces. We do not assume Lipschitz continuity of the nonlinearity, as previously done in the literature. First or second order convergence is derived, depending on the regularity of the solution, and a shortened proof for $o(1)$-convergence is given when only a mild solution exits. The analysis is also extended to the Lie scheme in a Banach space framework. The convergence results are illustrated by numerical experiments for Caginalp's solidification model and the Gray--Scott pattern formation problem.