A High-Order Compact (HOC) Implicit Difference Scheme and a Multigrid Method for Solving 3D Unsteady Reaction Diffusion Equations

Lili Wu,Xiufang Feng

doi:10.3390/math7121208

Abstract

A high-order compact (HOC) implicit difference scheme is proposed for solving three-dimensional (3D) unsteady reaction diffusion equations. To discretize the spatial second-order derivatives, the fourth-order compact difference operators are used, and the third- and fourth-order derivative terms, which appear in the truncation error term, are also discretized by the compact difference method. For the temporal discretization, the multistep backward Euler formula is used to obtain the fourth-order accuracy, which matches the spatial accuracy order. To accelerate the traditional relaxation methods, a multigrid method is employed, and the computational efficiency is greatly improved. Numerical experiments are carried out to validate the accuracy and efficiency of the present method.

Highlights

Reaction diffusion systems are widely used to describe the similar diffusion of all kinds of phenomena in modern science, such as the diffusion of gas, the infiltration of liquid concentration, gene diffusion in population, etc
This paper aims to develop an high-order compact (HOC) difference scheme combined with the multigrid method to solve the 3D unsteady reaction diffusion equations; an unconditionally stable and high-order five-level fully implicit scheme is proposed
A fully implicit five-layer high-order compact difference scheme is proposed for solving the 3D unsteady reaction diffusion equations

Summary

Introduction

Reaction diffusion systems are widely used to describe the similar diffusion of all kinds of phenomena in modern science, such as the diffusion of gas, the infiltration of liquid concentration, gene diffusion in population, etc. Most of these systems contain nonlinear reaction terms, which make it difficult to obtain the exact solutions [1,2]. Many researchers have been focusing on using efficient numerical methods for solving reaction diffusion equations. Wang and Guo [3] proposed a spatially fourth-order and temporally second-order accuracy implicit finite difference monotone scheme for solving nonlinear reaction diffusion equations. Araújo et al [4]

Objectives

Methods

Conclusion