An exponential integrator scheme for time discretization of nonlinear stochastic wave equation

Abstract: This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation. A unified framework is first set forth, which covers important cases of additive and multiplicative noises. Within this framework, the proposed scheme is shown to converge uniformly in the strong $L<sup>p$-sense with precise convergence rates given. The abstract results are then applied to several concrete examples. Further, weak convergence rates of the scheme are examined for the case of additive noise. To analyze the weak error for the nonlinear case, techniques based on the Malliavin calculus were usually exploited in the literature. Under certain appropriate assumptions on the nonlinearity, this paper provides a weak error analysis, which does not rely on the Malliavin calculus. The rates of weak convergence can, as expected, be improved in comparison with the strong rates. Both strong and weak convergence results obtained here show that the proposed method achieves higher convergence rates than the implicit Euler and Crank-Nicolson time discretizations. Numerical results are finally reported to confirm our theoretical findings.