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Quenching has been an extremely important natural phenomenon observed in many biomedical and multiphysical procedures, such as a rapid cancer cell progression or internal combustion process. The latter has been playing a crucial rule in optimizations of modern solid fuel rocket engine designs. Mathematically, quenching means the blowup of temporal derivatives of the solution function q while the function itself remains to be bounded throughout the underlying procedure. This paper studies a semi-adaptive numerical method for simulating solutions of a singular partial differential equation that models a significant number of quenching data streams. Numerical convergence will be investigated as well as verifying that features of the solution is preserved in the approximation. Orders of the convergence will also be validated through experimental procedures. Milne’s device will be used. Highly accurate data models will be presented to illustrate theoretical predictions. 

Nonlinear Kawarada equations have been used to model solid fuel combustion processes in the oil industry. An effective way to approximate solutions of such equations is to take advantage of the finite difference configurations. Traditionally, the nonlinear term of the equation is linearized while the numerical stability of a difference scheme is investigated. This leaves certain ambiguity and uncertainty in the analysis. Based on nonuniform grids generated through a quenching-seeking moving mesh method in space and adaptation in time, this paper introduces a completely new stability analysis of the approximation without freezing the nonlinearity involved. Pointwise orders of convergence are investigated numerically. Simulation experiments are carried out to accompany the mathematical analysis to strengthen our conclusions.