Approximation of Invariant Measure for Damped Stochastic Schr？dinger Equation via Ergodic Full Discretization

We study the approximation of invariant measure for an ergodic damped stochastic nonlinear Schr\"{o}dinger equation (NLSE) with additive noise. A spatial semi-discrete scheme and a fully discrete scheme for the damped stochastic NLSE are proposed. The ergodicity of the numerical solutions of both spatial semi-discretization and full discretization are proved. Also, we show that the approximation errors of invariant measure are $N^{-1}$ in spatial direction and $\tau^{\frac12}$ in temporal direction.