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

TL;DR

This paper introduces a nonlinear operator splitting algorithm that preserves positivity and monotonicity in numerical solutions of quenching combustion PDEs, ensuring qualitative fidelity to the original models.

Contribution

It presents a novel implicit nonlinear operator splitting method that maintains key qualitative features of quenching combustion PDEs, with rigorous proofs and convergence analysis.

Findings

Algorithm preserves positivity and monotonicity.

Convergence of the method is rigorously proven.

Explicit dependence on singularity is quantified.

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.