Splitting methods for constrained diffusion-reaction systems

TL;DR

This paper investigates Lie and Strang splitting methods for constrained diffusion-reaction systems, addressing order reduction issues in Strang splitting by introducing a correction term, and demonstrates improved convergence through numerical examples.

Contribution

It introduces a correction term to mitigate order reduction in Strang splitting for constrained PDEs, enhancing computational efficiency and accuracy.

Findings

Strang splitting suffers from order reduction due to inconsistent initial values.

Adding a correction term resolves order reduction without extra computational cost.

Numerical examples confirm the improved convergence of the proposed method.

Abstract

We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure seems promising as we can treat the nonlinearity separately. This has some computational advantages, since we only have to solve a linear constrained system and a nonlinear ODE. However, Strang splitting suffers from order reduction which limits its efficiency. This is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. The incorporation of an additional correction term resolves this problem without increasing the computational cost. Numerical examples including a coupled mechanical system illustrate the proven convergence results.