On numerical solutions of the stochastic wave equation

We show that there is a numerical scheme for the stochastic wave equation which converges in Lᵖ at a rate of O(√ h), and which converges a.s. uniformly on compact sets at a rate O(√ h|log h|⁾\,, for any >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.