高维非线性随机对流扩散方程的全离散LDG-IMEX方法的高阶矩稳定性与误差分析

中文摘要 AI 辅助

针对一类由乘性$\mathcal Q$-Wiener噪声驱动的高维非线性随机对流扩散方程，提出并分析了一种将局部间断伽辽金（LDG）方法与隐式-显式（IMEX）Euler时间离散相结合的全离散格式。该模型允许非线性首项系数、非线性对流项、耗散源项以及依赖梯度的噪声。扩散算子通过LDG公式隐式处理，而非线性对流、低阶漂移和随机项则显式评估。主要贡献在于对全离散格式进行了高阶矩稳定性与误差分析。一个核心困难在于非线性项导致路径相关的增长因子无法在整个样本空间上一致控制。为提供稳定性与误差估计，我们引入了递归定义的、适应数值解的嵌套子集。在所述随机抛物性和细化条件下，我们证明了这些子集的概率收敛到1。在这些子集上，数值解满足高阶矩稳定性，且全离散误差在空间上以任意接近$r+1$的阶收敛，在时间上以$1/2$阶收敛。我们还通过结合高阶矩误差界与离散Kolmogorov论证，推导了路径wise误差估计。针对随机Burgers方程和Allen-Cahn方程的数值实验证实了理论收敛阶，并展示了所提方法对非线性随机模型的鲁棒性。

英文摘要

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.

发表机构

• The Ohio State University（俄亥俄州立大学）
• Fudan University（复旦大学）

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