拟线性随机对流扩散型方程的全离散局部间断伽辽金方法

中文摘要 AI 辅助

本文针对一类由乘性 $\mathcal Q$-Wiener 噪声驱动的多维拟线性随机对流扩散型方程，提出并分析了一种采用 IMEX-Euler 时间离散的全离散局部间断伽辽金（LDG）方法。主扩散矩阵可能依赖于解以及空间和时间变量，而低阶漂移和噪声系数可能同时依赖于解及其梯度。在适当的随机抛物性条件下，我们建立了拟线性框架下全离散格式的无条件高阶矩稳定性估计。在半线性框架下，即主扩散矩阵不依赖于解但可能在空间和时间上变化时，我们进一步证明了空间阶为 $\mathcal O(h^{r+1})$、时间阶为 $\mathcal O(k^{1/2})$ 的最优高阶矩强误差估计。随后，通过将高阶矩误差界与离散 Kolmogorov 论证相结合，推导出逐路径误差估计。数值实验展示了所提方法的稳定性和收敛性。

英文摘要

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.

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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