On numerical solutions of the stochastic wave equation

We show that there is a numerical scheme for the stochastic wave equation which converges in L-p at a rate of O(root h), and which converges a.s. uniformly on compact sets at a rate O(root h\ log h\(epsilon)), for any epsilon > 0, where h is the step size in both time and space. We show that this is the optimal rate: there is no scheme depending on the same increments of white noise which has a higher order of convergence.