Parareal exponential $θ$-scheme for longtime simulation of stochastic Schrödinger equations with weak damping

Abstract: A parareal algorithm based on an exponential $\theta$-scheme is proposed for the stochastic Schr\"odinger equation with weak damping and additive noise. It proceeds as a two-level temporal parallelizable integrator with the exponential $\theta$-scheme as the propagator on the coarse grid. The proposed algorithm in the linear case increases the convergence order from one to $k$ for $\theta\in[0,1]\setminus{\frac12}$. In particular, the convergence order increases to $2k$ when $\theta=\frac12$ due to the symmetry of the algorithm. Furthermore, the algorithm is proved to be suitable for longtime simulation based on the analysis of the invariant distributions for the exponential $\theta$-scheme. The convergence condition for longtime simulation is also established for the proposed algorithm in the nonlinear case, which indicates the superiority of implicit schemes. Numerical experiments are dedicated to illustrate the best choice of the iteration number $k$, as well as the convergence order of the algorithm for different choices of $\theta$.