Convergence and error analysis of a semi-implicit finite volume scheme for the Gray--Scott system

TL;DR

This paper presents a detailed analysis of a semi-implicit finite volume scheme for the Gray--Scott system, proving its stability, convergence, and error estimates, supported by numerical experiments demonstrating pattern formation and convergence rate.

Contribution

It provides the first rigorous proof of convergence and error estimates for this scheme applied to the Gray--Scott system, including positivity and boundedness properties.

Findings

Proved unconditional well-posedness of the scheme.

Established strong convergence to weak solutions.

Validated a convergence rate of order 1 through numerical experiments.

Abstract

We analyze a semi-implicit finite volume scheme for the Gray--Scott system, a model for pattern formation in chemical and biological media. We prove unconditional well-posedness of the fully discrete problem and establish qualitative properties, including positivity and boundedness of the numerical solution. A convergence result is obtained by compactness arguments, showing that the discrete approximations converge strongly to a weak solution of the continuous system. Under additional regularity assumptions, we further derive a priori error estimates in the $L^2$ norm. Numerical experiments validate the theoretical analysis, confirm a rate of convergence of order 1, and illustrate the ability of the scheme to capture classical Gray--Scott patterns.