A positivity- and monotonicity-preserving nonlinear operator splitting approach for approximating solutions to quenching-combustion semilinear partial differential equations

Abstract

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.