Fully implicit time-stepping schemes and non-linear solvers for systems of reaction-diffusion equations

TL;DR

This paper develops and compares fully implicit time-stepping schemes and nonlinear solvers for reaction-diffusion systems, demonstrating that a single Newton iteration per timestep can achieve high accuracy efficiently, especially on complex evolving domains.

Contribution

It introduces the use of fractional-step theta methods with a single Newton iteration, showing they are as accurate as adaptive Newton methods and outperform Picard iteration for nonlinear reaction-diffusion equations.

Findings

Fractional-step theta method with one Newton iteration matches adaptive Newton accuracy.

Single Newton iteration is sufficient for high accuracy in complex nonlinear PDEs.

Numerical experiments validate theoretical advantages on stationary domains and surfaces.

Abstract

In this article we present robust, efficient and accurate fully implicit time-stepping schemes and nonlinear solvers for systems of reaction-diffusion equations. The applications of reaction-diffusion systems is abundant in the literature, from modelling pattern formation in developmental biology to cancer research, wound healing, tissue and bone regeneration and cell motility. Therefore, it is crucial that modellers, analysts and biologists are able to solve accurately and efficiently systems of highly nonlinear parabolic partial differential equations on complex stationary and sometimes continuously evolving domains and surfaces. The main contribution of our paper is the study of fully implicit schemes by use of the Newton method and the Picard iteration applied to the backward Euler, the Crank-Nicolson (and its modifications) and the fractional-step theta methods. Our results conclude that the fractional-step theta method coupled with a single Newton iteration at each timestep is as accurate as the fully adaptive Newton method; and both outperform the Picard iteration. In particular, the results strongly support the observation that a single Newton iteration is sufficient to yield as accurate results as those obtained by use of an adaptive Newton method. This is particularly advantageous when solving highly complex nonlinear partial differential equations on evolving domains and surfaces. To validate our theoretical results, various appropriate numerical experiments are exhibited on stationary planary domains and in the bulk of stationary surfaces.