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.
翻译:近年来,对数字算法的兴趣大增,这些算法保留了原有连续问题的各种质量特征。在这里,我们提议并调查一种数字算法,保留了所谓 " 灭火燃烧部分差异方程 " (PDEs)的质量特征。这种PDE常常用于模拟固体燃料点火过程或酶化学反应,其特点是其单数非线性反应术语及其存在时间间隔所显示的假设性和单一性解决方案。在本条中,我们提议了一种隐含的非线性操作者分离法,允许自然保存这些特征。算法的假设性和单一性得到了严格证明。此外,对算法的趋同性分析已经展开,对独一性的明确依赖在非线性环境下得到量化。