# On a Nonuniform Crank-Nicolson Scheme for Solving the Stochastic Kawarada Equation via Arbitrary Grids

## TL;DR

This paper introduces a nonuniform finite difference scheme on arbitrary grids for the stochastic Kawarada equation, ensuring stability, positivity, and monotonicity, with numerical experiments validating the approach.

## Contribution

It presents a novel nonuniform finite difference method on adaptive grids for the stochastic Kawarada equation, addressing singularities and uncertainty.

## Key findings

- Scheme preserves positivity and monotonicity under mesh constraints
- Numerical experiments confirm stability and accuracy
- Adaptive grids effectively handle singularities and stochastic sources

## 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.