Weak approximation of the stochastic wave equation

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