  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 $1$.
