  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 $L^p$-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.
