A splitting semi-implicit Euler method for stochastic incompressible Euler equations on 𝕋2

Abstract

Abstract The main difficulty in studying numerical methods for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (Printems, 2001, ESAIM Math. Model. Numer. Anal.35, 1055–1078). Although fruitful results on numerical approximations have been developed for SEEs, as far as we know, none of them include that of stochastic incompressible Euler equations. To bridge this gap, this paper proposes and analyzes a splitting semi-implicit Euler method in temporal direction for stochastic incompressible Euler equations on torus $\mathbb {T}^2$ driven by additive noises. 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 $1$.