This paper studies the numerical solution of a two-dimensional quenching type nonlinear reaction-diffusion problem via dimensional splitting. The variable coefficient differential equation considered possesses a nonlinear forcing term, and may lead to strong quenching singularities that have profound multiphysics and engineering applications to the energy industry. Our investigations focus on the construction and preservative properties of a semi-adaptive Peaceman-Rachford procedure for solving the aforementioned problem. In the study, the multidimensional differential equation is decomposed into single dimensional subequations so that the computational efficiency can be effectively raised. Temporal adaptation is implemented through arc-length estimations of the rate-of-change of the numerical solution. The positivity, monotonicity and localized linear stability of the variable step splitting scheme are analyzed. These ensure key preservations of underlying physical features of the computed approximations for applications. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and accuracy of the splitting method for solving singular quenching-combustion problems.

本文研究了二维猝灭型非线性反应扩散问题的维数分裂数值解。所考虑的变系数微分方程具有非线性强迫项，可能导致强猝灭奇点，在能源工业中具有深远的多物理场和工程应用。我们的研究集中在用于解决上述问题的半自适应 Peaceman-Rachford 过程的构造和保存特性。研究中将多维微分方程分解为单维子方程，有效提高了计算效率。时间适应是通过对数值解的变化率的弧长估计来实现的。积极性，分析了变步长分裂方案的单调性和局部线性稳定性。这些确保了应用程序计算近似值的基本物理特征的关键保留。提供了计算实验来说明我们的结果，并证明分裂方法解决奇异熄火-燃烧问题的可行性和准确性。