# A Time-Split MacCormack Scheme for Two-Dimensional Nonlinear Reaction-Diffusion Equations

## TL;DR

This paper introduces a three-level explicit time-split MacCormack scheme for efficiently solving two-dimensional nonlinear reaction-diffusion equations, demonstrating stability, convergence, and improved computational cost.

## Contribution

It presents a novel time-split MacCormack scheme with proven stability and convergence for nonlinear reaction-diffusion equations, reducing computational costs.

## Key findings

- The scheme is stable under CFL conditions.
- Convergence rate is validated through numerical experiments.
- The method reduces computational costs compared to traditional schemes.

## Abstract

A three-level explicit time-split MacCormack scheme is proposed for solving the two-dimensional nonlinear reaction-diffusion equations. The computational cost is reduced thank to the splitting and the explicit MacCormack scheme. Under the well known condition of Courant-Friedrich-Lewy (CFL) for stability of explicit numerical schemes applied to linear parabolic partial differential equations, we prove the stability and convergence of the method in $L^{\infty}(0,T;L^{2})$-norm. A wide set of numerical evidences which provide the convergence rate of the new algorithm are presented and critically discussed.