On a Nonuniform Crank-Nicolson Scheme for Solving the Stochastic Kawarada Equation via Arbitrary Grids
Joshua L Padgett, Qin Sheng
TL;DR
The paper tackles the degenerate stochastic Kawarada equation with a quenching singularity by developing a nonuniform, semi-adaptive Crank-Nicolson scheme on arbitrary space-time grids. By transforming to a scaled domain and discretizing space nonuniformly, it derives a semi-discrete system $v'(t)=Mv(t)+g(v(t))$ and applies a [1/1] Padé-based Crank-Nicolson time step, with a practical linearization of the nonlinear term. The authors prove positivity, monotonicity, and stability under mild mesh constraints and perform linear and semi-linear stability analyses to support robustness before quenching. Numerical experiments in 1D and 2D show how domain size, degeneracy, and stochastic forcing affect quenching times and locations, validating the method’s ability to capture sharp quenching events on arbitrary grids. The results suggest broad applicability to multidimensional Kawarada problems and outline future work on more advanced time-stepping, stochastic inhomogeneities, and fractional dynamics.
Abstract
This paper studies a nonuniform finite difference method for solving the degenerate Kawarada quenching-combustion equation with a vibrant stochastic source. Arbitrary grids are introduced in both space and time via adaptive principals to accommodate the uncertainty and singularities involved. It is shown that, under proper constraints on mesh step sizes, the positivity, monotonicity of the solution, and numerical stability of the scheme developed are well preserved. Numerical experiments are given to illustrate our conclusions.