A new approach on the stability and convergence of a time-space nonuniform finite difference approximation of a degenerate Kawarada problem

Abstract

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.

Keywords:

• Kawarada problem
• quenching solution
• finite difference method
• nonuniform meshes
• convergence
• stability

Mathematics Subject Classifications:

• 0M25
• 65C30
• 65M12
• 65M22
• 65N12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Both authors would like to thank the editor and anonymous reviewers for their time spent and the invaluable comments given. The suggestions have significantly improved the quality and presentation of this paper. Both authors would also appreciate the National Science Foundation USA, and Simons Foundation USA for generous support through [grant number DMS-2318032; NSF] and [grant number MPS-1001466; SIMONS].