Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise

Abstract

We introduce a fully discrete numerical scheme for semilinear SPDEs with additive space-time white noise that overcomes the previous order barrier for the spatial convergence rate. The scheme achieves a strong convergence rate of $M^{-1+\epsilon }$ in time and $N^{-3/2+\epsilon }$ in space for any $\epsilon >0$, where $M^{-1}$ and $N^{-1}$ are the temporal, respectively the spatial, meshsizes. This substantially improves the standard spatial error bounds of order $N^{-1/2}$ in the literature.

61 pages