标题： 随机Navier-Stokes方程的分裂法 显示英文标题

标题： Splitting Method for Stochastic Navier-Stokes Equations

摘要： 本文研究了带有加性随机噪声的二维随机定态Navier-Stokes（NS）方程。我们引入了一种创新的分裂方法，将随机NS方程分解为确定性的NS部分和一个随机方程。从等价性、稳定性、解的存在性和唯一性等方面严格分析了所提出的分裂方法。我们还提出了一种改进的分裂格式，通过省略其非线性项简化了随机方程，并提供了该改进方法解性质的详细分析。此外，我们讨论了原始分裂格式和改进方案的统计误差。我们的理论和数值研究表明，与原始随机NS方程相比，等价分裂方案表现出显著增强的稳定性，从而更有效地处理非线性特性。进行了若干数值实验以比较分裂方法和改进分裂方法的统计误差。值得注意的是，分裂方法中的确定性NS方程不需要重复求解，而改进方案中的随机方程没有非线性项。这些特性使改进的分裂方法在大规模计算中具有特别的优势，因为它显著提高了计算效率而不牺牲准确性。 显示英文摘要

摘要： This paper investigates the two-dimensional stochastic steady-state Navier-Stokes(NS) equations with additive random noise. We introduce an innovative splitting method that decomposes the stochastic NS equations into a deterministic NS component and a stochastic equation. We rigorously analyze the proposed splitting method from the perspectives of equivalence, stability, existence and uniqueness of the solution. We also propose a modified splitting scheme, which simplified the stochastic equation by omitting its nonlinear terms. A detailed analysis of the solution properties for this modified approach is provided. Additionally, we discuss the statistical errors with both the original splitting format and the modified scheme. Our theoretical and numerical studies demonstrate that the equivalent splitting scheme exhibits significantly enhanced stability compared to the original stochastic NS equations, enabling more effective handling of nonlinear characteristics. Several numerical experiments were performed to compare the statistical errors of the splitting method and the modified splitting method. Notably, the deterministic NS equation in the splitting method does not require repeated solving, and the stochastic equation in the modified scheme is free of nonlinear terms. These features make the modified splitting method particularly advantageous for large-scale computations, as it significantly improves computational efficiency without compromising accuracy.