A semi-adaptive compact splitting method for the numerical solution of 2-dimensional quenching problems

This article studies a semi-adaptive compact Peaceman–Rachford splitting method for solving two-dimensional nonlinear reaction–diffusion equations with singular source terms. While an adaption is utilized in the temporal direction, uniform grids are considered in the space. It is shown that the compact scheme is stable and convergent when its dimensional Courant numbers are within the frame of a window determined by the given spatial domain. Though such a window implies a considerable restriction on the decomposed compact scheme, the new computational strategy acquired is highly efficient and reliable for a variety of applications. Numerical examples are given to illustrate our conclusions and indicate that the method constructed is effective in determining key singularity characteristics such as the quenching time and critical domain.