Strong convergence rates for a full discretization of stochastic wave equation with nonlinear damping

Abstract

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 $\frac{1}{2}$ 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.

References in corpus (12)

• Galerkin approximations for the stochastic Burgers equation
• A trigonometric method for the linear stochastic wave equation
• Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noise
• An exponential integrator scheme for time discretization of nonlinear stochastic wave equation
• On the backward Euler approximation of the stochastic Allen-Cahn equation
• Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise
• Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise
• Error estimates of finite element method for semi-linear stochastic strongly damped wave equation
• An accelerated exponential time integrator for semi-linear stochastic strongly damped wave equation with additive noise
• Numerical approximation and simulation of the stochastic wave equation on the sphere
• Strong convergence rates of an explicit scheme for stochastic Cahn--Hilliard equation with additive noise
• Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs