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In recent years, there has been a large increase in interest in numericalalgorithms which preserve various qualitative features of the originalcontinuous problem. Herein, we propose and investigate a numerical algorithmwhich preserves qualitative features of so-called quenching combustion partialdifferential equations (PDEs). Such PDEs are often used to model solid-fuelignition processes or enzymatic chemical reactions and are characterized bytheir singular nonlinear reaction terms and the exhibited positivity andmonotonicity of their solutions on their time intervals of existence. In thisarticle, we propose an implicit nonlinear operator splitting algorithm whichallows for the natural preservation of these features. The positivity andmonotonicity of the algorithm is rigorously proven. Furthermore, theconvergence analysis of the algorithm is carried out and the explicitdependence on the singularity is quantified in a nonlinear setting.

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