Higher Order Pathwise Numerical Approximations of SPDEs with Additive Noise

In this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise is considered. A new numerical scheme for the time and space discretization of such SPDEs is proposed, and this scheme is shown to converge for such SPDEs faster than standard numerical schemes such as the linear implicit Euler scheme or the linear-implicit Crank–Nicholson scheme. The suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the SPDE. The abstract setting under which the scheme is analyzed includes SPDEs driven by fractional Brownian motions too.