An adaptive splitting approach for the quenching solution of reaction–diffusion equations over nonuniform grids

The numerical solution of a nonlinear degenerate reaction–diffusion equation of the quenching type is investigated. While spatial derivatives are discretized over symmetric nonuniform meshes, a Peaceman–Rachford splitting method is employed to advance solutions of the semidiscretized system. The temporal step is determined adaptively through a suitable arc-length monitor function. A criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution. Weak stability is proven in a von Neumann sense via the ∞∞-norm. Computational examples are presented to illustrate our results.