Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term

Xiaoliang Zhu,Yongbin Ge

doi:10.1155/2020/8161804

Abstract

An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the first-order and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.

Highlights

Nonlinear reaction-diffusion equation with a singular or near-singular source term has been widely applied in ion conduct polarization theory [1], computational fluid dynamics [2], electromagnetism [3], material research [4], ecology [5], thermology [6, 7], and so on
Blow-up and quenching are considered as singularity of a solution. The former indicates that its solution tends to infinite, and the latter indicates that its temporal derivative tends to infinite, but its solution is restricted in a certain scope [8, 9]
Ge et al firstly rendered a highorder compact difference scheme based on an adaptive mesh to study the quenching principle of the one-dimensional singular degenerate reaction-diffusion equation in 2018 [18]

Summary

Introduction

Nonlinear reaction-diffusion equation with a singular or near-singular source term has been widely applied in ion conduct polarization theory [1], computational fluid dynamics [2], electromagnetism [3], material research [4], ecology [5], thermology [6, 7], and so on. Ge et al firstly rendered a highorder compact difference scheme based on an adaptive mesh to study the quenching principle of the one-dimensional singular degenerate reaction-diffusion equation in 2018 [18]. A high-order adaptive difference scheme is firstly used to study the influence of degenerate function, convection function, nonlinear source function, and spatial definition length on quench behaviors for the one-dimensional convection-reaction-diffusion equation, respectively [23]. It provides the basis for adopting the high-precision algorithm to analyse the quenching states of the 2D convection-reaction-diffusion equation.

Semilinear Reaction-Diffusion Equation with Convection

Adaptive Mesh Method

Numerical Illustration

Conclusions