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We introduce a time-integrator to sample with high order of accuracy theinvariant distribution for a class of semilinear SPDEs driven by an additivespace-time noise. Combined with a postprocessor, the new method is amodification with negligible overhead of the standard linearized implicitEuler-Maruyama method. We first provide an analysis of the integrator whenapplied for SDEs (finite dimension), where we prove that the method has order$2$ for the approximation of the invariant distribution, instead of $1$. Wethen perform a stability analysis of the integrator in the semilinear SPDEcontext, and we prove in a linear case that a higher order of convergence isachieved. Numerical experiments, including the semilinear heat equation drivenby space-time white noise, confirm the theoretical findings and illustrate theefficiency of the approach.

High-order integrator for sampling the invariant distribution of a class of parabolic SPDEs with additive space-time noise