Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

Abstract

Abstract.A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Störmer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.

Drift-preserving numerical integrators for stochastic Hamiltonian systems

“…This is not a problem for polynomial potentials V for instance. We also observe that, as it happens for deterministic energy-preserving methods, the numerical scheme (3) only requires the evaluation to the vector field of problem (1) in selected points, without requiring additional projection steps for the exact conservation of the trace formula (2), that would inflate the computational cost of the procedure.…”

Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

“…Applications of such numerical schemes to the deterministic (nonlinear) Schrödinger equation can be found in, for example, [4][5][6][7][8][9][10]17,21] and references therein. Furthermore, these numerical methods were investigated for stochastic parabolic partial differential equations in, for example, [23][24][25], more recently for the stochastic wave equations in [2,11,12,27], where they are termed stochastic trigonometric methods, and lately to stochastic Schrödinger equations driven by Ito noise in [1].…”

An accelerated exponential time integrator for semi-linear stochastic strongly damped wave equation with additive noise

“…Different from the usual Euler-typr time-stepping schemes using the basic increments of the driven Wiener process [1,3,13,15,19,20,24,27], the accelerated exponential time integrators rely on suitable linear functionals of the Wiener process and usually attain superconvergence rates in time [7,8,25,26]. In 2009, such scheme was first constructed by Jentzen and Kloeden [8] for semi-linear parabolic SPDEs with additive space-time white noise.…”

“…Afterwards, the accelerated schemes were extended to solve a larger class of parabolic SPDEs with more general noise and the error bounds were analyzed under relaxed conditions on the nonlinearity [7,26]. Furthermore, the accelerated scheme was successfully adapted to solve semilinear stochastic wave equations and the order barrier 1 2 was went beyond [25]. Following the idea of the acceleration technique, we discretize the considered problem (1.1) in space by a spectral Galerkin method and in time by an exponential integrator involving linear functionals of the noise.…”