Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

TL;DR

This paper provides a comprehensive convergence analysis and error estimates for an energy-stable operator splitting scheme applied to a reaction-diffusion system with detailed balance, addressing nonlinear and singular terms.

Contribution

It offers the first detailed convergence proof for a variational operator splitting scheme with energy stability for reaction-diffusion systems with detailed balance.

Findings

The scheme is energy stable and positivity-preserving.

Convergence and error bounds are established in the discrete maximum norm.

The analysis handles nonlinear and singular logarithmic reaction terms effectively.

Abstract

We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu et al.,J. Comput. Phys., 436, 110253, 2021] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms, which are treated implicitly in the chemical reaction stage. In addition, a combination of a rough error estimate and a refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting results in the convergence estimate of the numerical scheme for the full reaction-diffusion system.