以 $\mathbb T $2 $ 分解的半隐含方法, 用于以 $\ mathbb T $2$ 的蒸气性不压缩 Euler 方程 (A splitting semi-implicit method for stochastic incompressible Euler equations on $\mathbb T^2$)

The main difficulty in studying numerical method for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (J. Printems. [ESAIM Math. Model. Numer. Anal. (2001)]). Although fruitful results on numerical approximations for SEEs have been developed, as far as we know, none of them include that of stochastic incompressible Euler equations. To bridge this gap, this paper proposes and analyses a splitting semi-implicit method in temporal direction for stochastic incompressible Euler equations on torus $\mathbb{T}^2$ driven by an additive noise. By a Galerkin approximation and the fixed point technique, we establish the unique solvability of the proposed method. Based on the regularity estimates of both exact and numerical solutions, we measure the error in $L^2(\mathbb{T}^2)$ and show that the pathwise convergence order is nearly $\frac{1}{2}$ and the convergence order in probability is almost $1$.

翻译：研究随机进化方程式(SEE)的数值方法的主要困难在于对时间离散的处理(J. Printems.[ESAIM Math.model.Numer.Anal.(2001)])。虽然就我们所知,在SEES的数值近似值方面已经取得了丰硕的成果,但其中没有一项包括Stochasteic 无法压缩的Euler方程式。为了缩小这一差距,本文件建议并分析一种分解的半隐含法,即对由添加性噪音驱动的torus\ $mathbb{T ⁇ 2$的可抑制性软化方程式,在时间方向上采用分解的半隐含法。通过加勒金近似法和固定点法,我们确定了拟议方法的独特可溶性。根据精确和数字解决方案的规律性估计,我们用$L2(\\\mathb{T ⁇ 2)来测量误差,并显示路径趋同值近$\frac{1 ⁇ 2}和近似合值为$1美元。