High-Moment Stability and Error Analysis of a Fully Discrete LDG-IMEX Method for High Dimensional Nonlinear Stochastic Convection-Diffusion Equations

Source abstract

A fully discrete local discontinuous Galerkin (LDG) method coupled with an implicit-explicit (IMEX) Euler time discretization is presented and analyzed for a class of high dimensional nonlinear stochastic convection-diffusion equations driven by multiplicative $\mathcal Q$-Wiener noise. The model allows nonlinear leading coefficients, nonlinear convection terms, dissipative source terms, and gradient-dependent noise. The diffusion operator is treated implicitly through the LDG formulation, while the nonlinear convection, lower-order drift, and stochastic terms are evaluated explicitly. The main contribution is a high-moment stability and error analysis for the fully discrete scheme. A central difficulty is that the nonlinear terms lead to pathwise growth factors that cannot be controlled uniformly on the full sample space. To provide the stability and error estimate, we introduce recursively defined nested subsets adapted to the numerical solution. Under the stated stochastic parabolicity and refinement conditions, we prove that these subsets have probabilities converging to one. On these subsets, the numerical solution satisfies high-moment stability, and the fully discrete error converges with order arbitrarily close to $r+1$ in space and $1/2$ in time. We also derive a pathwise error estimate by combining the high-moment error bound with a discrete Kolmogorov argument. Numerical experiments for stochastic Burgers' and Allen-Cahn equations confirm the theoretical rates and demonstrate the robustness of the proposed method for nonlinear stochastic models.