Optimal error estimates and stability of a local discontinuous Galerkin method for the stochastic two-dimensional KdV equation

The stochastic two-dimensional KdV equation arises as a mathematical model for shallow water wave dynamics in physical systems. To efficiently handle the equation’s high-order spatial derivatives and stochastic terms, a local discontinuous Galerkin method is proposed. The method is proved to be L 2 -stable and to achieve the optimal mean-square convergence rate of order N + 1 when degree- N polynomials are used. For temporal discretization, the implicit midpoint method is applied, and the restarted Generalized Minimum Residual method is employed to solve the resulting linear systems in two-dimensional simulations. Numerical experiments demonstrate optimal convergence rates and confirm both the theoretical analysis and the effectiveness of the method.