Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition

In this paper we obtain a general statement concerning pathwise convergence of the full discretization of certain stochastic partial differential equations (SPDEs) with non-globally Lipschitz continuous drift coefficients. We focus on non-diagonal colored noise instead of the usual space-time white noise. By applying a spectral Galerkin method for spatial discretization and a numerical scheme in time introduced by Jentzen, Kloeden and Winkel we obtain the rate of path-wise convergence in the uniform topology. The main assumptions are either uniform bounds on the spectral Galerkin approximation or uniform bounds on the numerical data. Numerical examples illustrate the theoretically predicted convergence rate.