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

Abstract

This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed by a spectral Galerkin method, we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise. Under appropriate assumptions, we provide error bounds for the proposed full-discrete scheme. It is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem, which allows for higher convergence rate than usual time-stepping schemes. For the space-time white noise case in two or three spatial dimensions, the scheme still exhibits a good convergence performance. Another striking finding is that, even for the velocity with low regularity the scheme always promises first order strong convergence in time. Numerical examples are finally reported to confirm our theoretical findings.

Strong convergence of full-discrete nonlinearity-truncated accelerated exponential euler-type approximations for stochastic Kuramoto–Sivashinsky equations

“…First, we note that the approximation scheme (4) does not temporally discretize the semigroup (e tA ) t∈[0,∞) appearing in (2) and is thus an appropriate modification of the accelerated exponential Euler scheme in Section 3 in Jentzen & Kloeden [21] (cf., e.g., also Section 4 in Jentzen & Kloeden [20] for an overview and e.g., Lord & Tambue [27] and Wang & Qi [35] for further results on accelerated exponential Euler approximations). This lack of discretization of the semigroup in the stochastic integral (2) has been proposed in Jentzen & Kloeden [21] to obtain an approximation scheme which converges under suitable assumptions with a significant higher convergence rate than previously analyzed approximation schemes such as the linear implicit Euler scheme or the exponential Euler scheme (cf., e.g., Theorem 3.1 in Jentzen & Kloeden [21], Theorem 1 in [22], Theorem 3.1 in Wang & Qi [35], and Theorem 3.1 in Qi & Wang [29]). In this article the lack of discretization of the semigroup in the non-stochastic integral in (2) is employed for a different purpose, that is, here this lack of discretization is used to obtain a scheme that inherits an appropriate a priori estimate from the exact solution process of the SPDE (3).…”

An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation

“…Since the stochastic convolution is Gaussian distributed and diagonalizable on {e i } i∈N , the scheme is much easier to simulate than it appears at first sight (see comments in section 5 for the implementation). When the nonlinearity grows super-linearly, one can in general not expect that the usual AEE schemes [25,26,35,39,45,46] converge strongly, based on the observation that the standard Euler method strongly diverges for ordinary (finite dimensional) SDEs [21]. Also, we mention that analyzing the strong convergence rate is much more difficult than that in the finite dimensional SDE setting.…”

“…Furthermore, we would like to point out that the improvement of convergence rate is essentially credited to fully preserving the stochastic convolution in the time-stepping scheme (4.1). Such a kind of accelerating technique is originally due to Jentzen and Kloeden [26], simulating nearly linear parabolic SPDEs and has been further examined and extended in different settings [25,35,39,45,46], where a globally Lipschitz condition imposed on nonlinearity is indispensable. When the nonlinearity grows super-linearly and the globally Lipschitz condition is thus violated, one can in general not expect the usual accelerated exponential time-stepping schemes converge in the strong sense, based on the observation that the standard Euler method strongly diverges for ordinary (finite dimensional) SDEs [21].…”

On strong convergence of a fully discrete scheme for solving stochastic strongly damped wave equations

“…There are many researches on the asymptotic behaviors of solutions and random attractors [6,29,30,35] for SSDWEs. However, only a few references consider numerical approximations to SSDWEs [20,21,27]. This paper attempts to establish a fully discrete scheme for the SSDWE by a spectral approximation of the noise and present its strong error estimates.…”