  In this paper, we present an energy-preserving exponentially integrable
numerical method for stochastic wave equation with cubic nonlinearity and
additive noise. We first apply the spectral Galerkin method to discretizing the
original equation and show that this spatial discretization possesses an energy
evolution law and certain exponential integrability property. Then the
exponential integrability property of the exact solution is deduced by proving
the strong convergence of the semi-discretization. To propose a full discrete
numerical method which could inherit both the energy evolution law and the
exponential integrability, we use the splitting technique and averaged vector
field method in the temporal direction. Combining these structure-preserving
properties with regularity estimates of the exact and the numerical solutions,
we obtain the strong convergence rate of the numerical method. Numerical
experiments coincide with these theoretical results.
