Exponential integrator for stochastic strongly damped wave equation based on the Wong–Zakai approximation

Abstract

We consider strong convergence of numerical approximations for a stochastic strongly damped wave equation driven by a class of additive space–time noises characterized by a parameter β∈(−1,2]. With the help of the Wong–Zakai (WZ) approximation to the noise, i.e., replacing the driven noise with its finite spectral expansion truncation, we obtain an approximate equation. Based on the consistency and high regularity of the approximate equation, we develop two exponential integrators for time-stepping discretization and use the spectral Galerkin method in space to develop full-discrete schemes. Performing error estimates in the strong sense, we show that the optimal strong order in time of the proposed WZ-approximation-based exponential Euler scheme is min{1,1+β/2−ϵ}, which is |β|/2 order higher than the order of min{1,1+β} in the existing works when β∈(−1,0). Moreover, we prove that the proposed WZ-approximation-based exponential trapezoidal scheme is of min{3/2,1+β/2−ϵ}-order strong convergence in time when β∈[−1/4,2], so it can break the first order barrier and obtain a higher accurate numerical solution. Numerical examples are performed to verify and develop the theoretical findings.