In the present work, strong approximation errors are analyzed for both the spatial semi-discretization and the spatio-temporal fully discretization of stochastic wave equations (SWEs) with cubic polynomial nonlinearities and additive noises. The fully discretization is achieved by the standard Galerkin ffnite element method in space and a novel exponential time integrator combined with the averaged vector ffeld approach. The newly proposed scheme is proved to exactly satisfy a trace formula based on an energy functional. Recovering the convergence rates of the scheme, however, meets essential difffculties, due to the lack of the global monotonicity condition. To overcome this issue, we derive the exponential integrability property of the considered numerical approximations, by the energy functional. Armed with these properties, we obtain the strong convergence rates of the approximations in both spatial and temporal direction. Finally, numerical results are presented to verify the previously theoretical findings.

翻译：本文针对具有立方多项式非线性项和加性噪声的随机波动方程（SWEs），分析了其空间半离散化及时空全离散化的强逼近误差。全离散化方案通过标准的Galerkin有限元方法进行空间离散，并结合一种新型指数时间积分器与平均向量场方法实现。研究证明，新提出的格式能精确满足基于能量泛函的迹公式。然而，由于缺乏全局单调性条件，恢复该格式的收敛阶面临本质困难。为克服此问题，我们借助能量泛函推导了所考虑数值逼近的指数可积性。基于这些性质，我们获得了逼近在空间和时间方向上的强收敛阶。最后，数值实验结果验证了先前的理论结论。