A second-order accurate, operator splitting scheme for reaction-diffusion systems in an energetic variational formulation

TL;DR

This paper introduces a second-order accurate, energy-stable operator splitting scheme for reaction-diffusion systems, ensuring positivity and energy dissipation, based on an energetic variational formulation and validated through numerical experiments.

Contribution

It develops a novel second-order, positivity-preserving, energy-stable operator splitting scheme for reaction-diffusion equations using an energetic variational approach.

Findings

The scheme is second-order accurate in time.

It preserves positivity and energy stability.

Numerical experiments confirm the scheme's accuracy.

Abstract

A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting numerical scheme is proposed and analyzed for the system of reaction-diffusion equations with detailed balance. The scheme is designed based on an 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. At the reaction stage, the reaction trajectory equation is approximated by a second-order Crank-Nicolson type method. The unique solvability, positivity-preserving, and energy-stability are established based on a convexity analysis. In the diffusion stage, an exact integrator is applied if the diffusion coefficients are constant, and a Crank-Nicolson type scheme is applied if the diffusion process becomes nonlinear. In either case, both the positivity-preserving property and energy stability could be theoretically established. Moreover, a combination of the numerical algorithms at both stages by the Strang splitting approach leads to a second-order accurate, structure-preserving scheme for the original reaction-diffusion system. Numerical experiments are presented, which demonstrate the accuracy of the proposed scheme.