Higher order time discretization for the stochastic semilinear wave equation with multiplicative noise

Abstract

In this paper, a higher order time-discretization scheme is proposed, where the iterates approximate the solution of the stochastic semilinear wave equation driven by multiplicative noise with general drift and diffusion. We employ variational method for its error analysis and prove an improved convergence order of 3 2 for the approximates of the solution. The core of the analysis is Hölder continuity in time and moment bounds for the solutions of the continuous and the discrete problem. Computational experiments are also presented.

Fully discrete finite element methods for nonlinear stochastic elastic wave equations with multiplicative noise

“…Like in the numerical analysis of stochastic parabolic PDEs, a key idea and technique for overcoming the difficulty caused by the noise is to establish and make use of the Hölder continuity in time of the weak solution in various spatial norms. We also note that the results of this paper could be improved in light of the recent work [18]. In particular, it is possible to construct better timestepping schemes which can achieve the optimal 3 2 order of convergence for the displacement approximation, which in turn requires higher order approximation for the multiplicative noise term.…”

“…The next theorem establishes the time semi-discrete L 2 -error of u n , and it states that the time semidiscrete convergence order in L 2 -norm is 1. Its proof is inspired by a similar proof for the stochastic scalar wave equation given in [18]. Theorem 3.5.…”

“…where u n, 1 2 := 1 2 u n+1 + u n−1 . However, such an improvement could not be realized unless some more involved higher order treatment of the noise term is adopted as demonstrated in [18] for the corresponding stochastic acoustic wave equations.…”

“…There is a large amount of literature on numerical methods for deterministic acoustic wave equations, we refer the reader to [1,3,4,5,7,8,16,17,21,22,29,32,33,35,38,40] and the references therein for a detailed account. Moreover, numerical methods for stochastic acoustic wave equations have also been intensively developed in the last few years, see [2,10,12,14,15,18,20,23,24,26,27,30,37]. Similarly, the elastic wave equations are also of great importance and find applications in geoscience for modeling seismic waves and in medical science for tumor detection as well as in materials science for non-destructive testing.…”