$\mathbb T^2$ 上随机不可压缩 Euler 方程的分裂半隐式方法
A splitting semi-implicit method for stochastic incompressible Euler equations on $\mathbb T^2$
AI总结：
本文针对环面 $\mathbb{T}^2$ 上由加性噪声驱动的随机不可压缩 Euler 方程，提出时间方向的分裂半隐式方法，利用 Galerkin 逼近与不动点技术证明唯一可解性，并得出逐路径与依概率的收敛阶。
摘要：
研究随机演化方程（SEEs）数值方法的主要困难在于时间离散化的处理（J. Printems. [ESAIM Math. Model. Numer. Anal. (2001)]）。尽管关于 SEEs 数值逼近的成果已很丰富，但据我们所知，尚无针对随机不可压缩 Euler 方程的研究。为填补此空白，本文针对受加性噪声驱动、定义在环面 $\mathbb{T}^2$ 上的随机不可压缩 Euler 方程，提出并分析了时间方向上的分裂半隐式方法。通过 Galerkin 逼近和不动点技术，我们建立了该方法的唯一可解性。基于精确解和数值解的正则性估计，我们在 $L^2(\mathbb{T}^2)$ 中度量误差，并证明其逐路径收敛阶接近 $\frac{1}{2}$，依概率收敛阶几乎为 $1$。
英文摘要：
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 $1$.
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