In recent years, there has been a large increase in interest in numerical algorithms which preserve various qualitative features of the original continuous problem. Herein, we propose and investigate a numerical algorithm which preserves qualitative features of so-called quenching combustion partial differential equations (PDEs). Such PDEs are often used to model solid-fuel ignition processes or enzymatic chemical reactions and are characterized by their singular nonlinear reaction terms and the exhibited positivity and monotonicity of their solutions on their time intervals of existence. In this article, we propose an implicit nonlinear operator splitting algorithm which allows for the natural preservation of these features. The positivity and monotonicity of the algorithm is rigorously proven. Furthermore, the convergence analysis of the algorithm is carried out and the explicit dependence on the singularity is quantified in a nonlinear setting.

中文翻译：

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一种用于逼近淬火燃烧半线性偏微分方程解的正性和单调性非线性算子分裂方法

近年来，对保留原始连续问题的各种定性特征的数值算法的兴趣大大增加。在此，我们提出并研究了一种数值算法，该算法保留了所谓的淬火燃烧偏微分方程 (PDE) 的定性特征。此类偏微分方程通常用于模拟固体燃料点火过程或酶化学反应，其特征在于其奇异的非线性反应项以及其解在其存在时间间隔内所表现出的正性和单调性。在本文中，我们提出了一种隐式非线性算子分裂算法，可以自然地保留这些特征。算法的积极性和单调性得到了严格的证明。此外，