The paper establishes the strong convergence rates of a spatio-temporal full discretization of the stochastic wave equation with nonlinear damping in dimension one and two. We discretize the SPDE by applying a spectral Galerkin method in space and a modified implicit exponential Euler scheme in time. The presence of the super-linearly growing damping in the underlying model brings challenges into the error analysis. To address these difficulties, we first achieve upper mean-square error bounds, and then obtain mean-square convergence rates of the considered numerical solution. This is done without requiring the moment bounds of the full approximations. The main result shows that, in dimension one, the scheme admits a convergence rate of order $\tfrac12$ in space and order $1$ in time. In dimension two, the error analysis is more subtle and can be done at the expense of an order reduction due to an infinitesimal factor. Numerical experiments are performed and confirm our theoretical findings.

翻译：本文建立了在二维和三维空间中具有非线性阻尼的随机波动方程时空全离散格式的强收敛速率。我们采用空间上的谱伽辽金方法和时间上的修正隐式指数欧拉格式对该随机偏微分方程进行离散化。模型中存在超线性增长的阻尼项给误差分析带来了挑战。为解决这些困难，我们首先获得均方误差上界，进而导出所考虑数值解的均方收敛速率，此过程无需逼近解矩有界条件。主要结果表明：在一维空间中，该格式具有空间方向$\tfrac12$阶和时间方向$1$阶的收敛速率；在二维空间中，误差分析更为复杂，需以无穷小因子导致的阶数降低为代价。数值实验验证了理论结果的正确性。