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

05/19/2021
by   Chun Liu, et al.
0

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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 13

10/30/2020

A Structure-preserving, Operator Splitting Scheme for Reaction-Diffusion Equations Involving the Law of Mass Action

In this paper, we propose and analyze a positivity-preserving, energy st...
09/07/2021

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

A second-order accurate in time, positivity-preserving, and unconditiona...
09/17/2020

A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system

In this paper we propose and analyze a finite difference numerical schem...
10/09/2019

Convergence analysis of a numerical scheme for the porous medium equation by an energetic variational approach

The porous medium equation (PME) is a typical nonlinear degenerate parab...
05/04/2022

Performance evaluations on the parallel CHAracteristic-Spectral-Mixed (CHASM) scheme

Performance evaluations on the deterministic algorithms for 6-D problems...
06/06/2021

Error estimate of a decoupled numerical scheme for the Cahn-Hilliard-Stokes-Darcy system

We analyze a fully discrete finite element numerical scheme for the Cahn...
05/09/2021

Fast stable finite difference schemes for nonlinear cross-diffusion

The dynamics of cross-diffusion models leads to a high computational com...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.