  The numerical solution of a highly nonlinear two-dimensional degenerate
stochastic Kawarada equation is investigated. A semi-discretized approximation
in space is comprised on arbitrary nonuniform grids. Exponential splitting
strategies are then applied to advance solutions of the semi-discretized scheme
over adaptive grids in time. It is shown that key quenching solution features
including the positivity and monotonicity are well preserved under modest
restrictions. The numerical stability of the underlying splitting method is
also maintained without any additional restriction. Computational experiments
are provided to not only illustrate our results, but also provide further
insights into the global nonlinear convergence of the numerical solution.
