Stability and Optimal Error Estimates Analysis of an LDG Method for the Stochastic Nonlinear KdV Equation

Abstract

To address the computational challenges of stochastic nonlinear partial differential equations with high-order derivatives, a local discontinuous Galerkin method is proposed for the stochastic KdV equation. The method is proven to be $\mathcal{L}^2$-stable and to attain optimal error estimates of order $n+1$ measured in the mean-square norm when degree-$n$ polynomials are used. Temporal integration of the spatial semi-discrete stochastic system in the numerical experiments is carried out by using the implicit midpoint method. The simulation results verify the method's accuracy and its consistency with the theoretical analysis.