Weak approximation of the stochastic wave equation

AbstractWe investigate the accuracy of approximation of E[φ(u(t))], where {u(t):t∈[0,∞)} is the solution of the stochastic wave equation driven by the space–time white noise and φ is an R-valued function defined on the Hilbert space L2(R). The approximation is done by the leap-frog scheme. We show that, under certain conditions on φ, the approximation by the leap-frog scheme is of order two

Analysis of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations

We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations driven by additive It\uf4 noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme