×

Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noise. (English) Zbl 1322.65023

Summary: Novel fully discrete schemes are developed to numerically approximate a semilinear stochastic wave equation driven by additive space-time white noise. A spectral Galerkin method is proposed for the spatial discretization, and exponential time integrators involving linear functionals of the noise are introduced for the temporal approximation. The resulting fully discrete schemes are very easy to implement and allow for a higher strong convergence rate in time than existing time-stepping schemes such as the Crank-Nicolson-Maruyama scheme and the stochastic trigonometric method. Particularly, it is shown that the new schemes achieve in time an order of \(1- \epsilon\) for arbitrarily small \(\epsilon >0\), which exceeds the barrier order \(\frac{1}{2}\) established by J. B. Walsh [Ill. J. Math. 50, No. 1–4, 991-1018 (2006; Zbl 1108.60058)]. Numerical results confirm higher convergence rates and computational efficiency of the new schemes.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65J15 Numerical solutions to equations with nonlinear operators

Citations:

Zbl 1108.60058
PDFBibTeX XMLCite
Full Text: DOI arXiv