A Fully Discrete Local Discontinuous Galerkin Method for Quasilinear Stochastic Convection-Diffusion-Type Equations

Abstract

In this paper, we develop and analyze a fully discrete local discontinuous Galerkin (LDG) method with IMEX-Euler time discretization for a class of multi-dimensional quasilinear stochastic convection-diffusion-type equations driven by multiplicative $\mathcal Q$-Wiener noise. The leading diffusion matrix may depend on the solution as well as the spatial and temporal variables, while the lower-order drift and noise coefficients may depend on both the solution and its gradient. Under a suitable stochastic parabolicity condition, we establish unconditional high-moment stability estimates for the fully discrete scheme in the quasilinear setting. In the semilinear setting, where the leading diffusion matrix is independent of the solution but may vary in space and time, we further prove optimal high-moment strong error estimates of order $\mathcal O(h^{r+1})$ in space and $\mathcal O(k^{1/2})$ in time. A pathwise error estimate is then derived by combining the high-moment error bound with a discrete Kolmogorov argument. Numerical experiments are presented to illustrate the stability and convergence properties of the proposed method.