A splitting semi-implicit method for stochastic incompressible Euler equations on $\mathbb T^2$

TL;DR

This paper introduces and analyzes a splitting semi-implicit numerical method for stochastic incompressible Euler equations on a 2D torus, demonstrating nearly 1/2 order pathwise convergence and almost 1 order in probability.

Contribution

It develops the first splitting semi-implicit scheme for stochastic incompressible Euler equations and proves its convergence properties.

Findings

Pathwise convergence order is nearly 1/2.

Convergence order in probability is almost 1.

Unique solvability of the proposed method is established.

Abstract

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 $1$.