一类半线性抛物型燃烧爆炸模型的数值解

数学理论与应用 ›› 2020, Vol. 40 ›› Issue (3): 40-53.

一类半线性抛物型燃烧爆炸模型的数值解

杨红宾1，徐宇锋2,*

1.南开大学数学科学学院，天津 300071;
2.中南大学数学与统计学院，长沙 410083

出版日期:2020-09-30 发布日期:2021-03-25
通讯作者: 徐宇锋(1987−)，男，湖南长沙人，副教授，博士，从事计算数学研究；E−mail：xuyufeng@csu.edu.cn
基金资助:
湖南省自然科学基金资助项目(2019JJ50755)

Numerical solution for a semilinear parabolic blowup-combustion model

Online:2020-09-30 Published:2021-03-25

摘要/Abstract

摘要： 研究了一类燃烧爆炸数学模型的有限差分解法. 方程的指数源项刻画了Arrhenius反应速率, 常用于模拟刚性点火过程. 引入截断函数, 原问题转化为具有一致有界连续解的近似模型. 当截断常数充分大时, 模型的爆破解和几乎完全爆破解可以通过求解近似系统获得, 且求解过程不产生数值爆炸. 适当优化步长选择, 爆破位置和爆破时刻的逼近效果比直接离散原问题求解更简单, 计算效果更好. 最后, 数值实验验证了理论分析的结果和模型的物理性质.

关键词: 半线性抛物型方程 , 有限差分法 , 指数源项 , 单调性 , 截断函数

Abstract: In this paper, we study an efficient finite difference method based on constructed mesh for a class of semilinear parabolic equation arising from combustion process. Due to the influence of exponential source term, the considered model may theoretically describe the rigid ignition process in combustion. To catch the blowup and almost complete blowup phenomena, we introduce a cut-off level constant in the indicator attached to source term, so that the blowup position and moment can be approximated with less effort. Finally, numerical simulations with various settings of parameters and initial conditions are given to illustrate the excellent performance of numerical scheme, as well as fruitful dynamics of the blowup-combustion model.

Key words: Semilinear parabolic equation;Finite difference method, Exponential source, Monotonicity; Cut-off level function

杨红宾, 徐宇锋. 一类半线性抛物型燃烧爆炸模型的数值解[J]. 数学理论与应用, 2020, 40(3): 40-53.