Strong Convergence of a Verlet Integrator for the Semilinear Stochastic Wave Equation

Banjai, Lehel; Lord, Gabriel J.; Molla, Jeta

doi:10.1137/20m1364746

Cited by 8 publications

(5 citation statements)

References 30 publications

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“…. It is not difficult to show that the energy of the system (1.1) evolutes linearly with a rate Tr(Q), i.e., E u(t) 2 V = E u 0 2 V + Tr(Q)t. Note that (2.2) is an infinite-dimensional Hamiltonian system. If the coefficients ε, µ are constants, the canonical form of the infinite-dimensional Hamiltonian system of (2.2) reads…”

Section: Hs(uu)mentioning

confidence: 99%

“…We discretize the temporal semidiscretization (3.1) further in space using a dG method, and then it results the fully discrete method (5.1), called the symplectic dG full discretization; see also Section 4 for the treatment of the dG approximation of stochastic Maxwell equations. We refer interested readers to [17] for the application of dG methods to the time-harmonic stochastic Maxwell equations with color noise, to [3] for the application to stochastic Helmholtz-type equation, to [1] for the application to stochastic Allen-Cahn equation, to [2] for the application to the semi-linear stochastic wave equation, to [14] for the application to stochastic conservation laws, and to [4] for the application of a symplectic local dG method to stochastic Schrödinger equation. Since the highest regularity of stochastic Maxwell equations that can be guaranteed is in H 2 , the dG space is taken to be the set of piecewise linear functions.…”

Section: Introductionmentioning

confidence: 99%

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A symplectic discontinuous Galerkin full discretization for stochastic Maxwell equations

Chen¹

2020

Preprint

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This paper proposes a fully discrete method called the symplectic dG full discretization for stochastic Maxwell equations driven by additive noises, based on a stochastic symplectic method in time and a discontinuous Galerkin (dG) method with the upwind fluxes in space.A priori H k -regularity (k ∈ {1, 2}) estimates for the solution of stochastic Maxwell equations are presented, which have not been reported before to the best of our knowledge. These H kregularities are vital to make the assumptions of the mean-square convergence analysis on the initial fields, the noise and the medium coefficients, but not on the solution itself. The convergence order of the symplectic dG full discretization is shown to be k/2 in the temporal direction and k − 1/2 in the spatial direction. Meanwhile we reveal the small noise asymptotic behaviors of the exact and numerical solutions via the large deviation principle, and show that the fully discrete method preserves the divergence relations in a weak sense.

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Section: Hs(uu)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A symplectic discontinuous Galerkin full discretization for stochastic Maxwell equations

Chen¹

2020

Preprint

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“…Besides global Lipschitz continuity, no further regularity assumptions are imposed on the nonlinearity F and noise G. Now, our goal is to show pathwise uniform convergence of contractive time discretisation schemes for such irregular nonlinearities and rough initial data, focusing on the hyperbolic setting. It has been extensively studied in recent years (see [1,2,4,7,[10][11][12][14][15][16][17]19,20,26,29,32,[36][37][38][39]42,53] and references therein). When passing to the parabolic setting (i.e.…”

Section: Introductionmentioning

confidence: 99%

Temporal approximation of stochastic evolution equations with irregular nonlinearities

Klioba,

Veraar

2024

J. Evol. Equ.

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In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error$$\begin{aligned} \textrm{E}_{k}^{\infty } {:}{=}\Big (\mathbb {E}\sup _{j\in \{0, \ldots , N_k\}} \Vert U(t_j) - U^j\Vert _X^p\Big )^{1/p} \rightarrow 0\quad (k \rightarrow 0), \end{aligned}$$ E k ∞ : = ( E sup j ∈ { 0 , … , N k } ‖ U ( t j ) - U j ‖ X p ) 1 / p → 0 ( k → 0 ) , where $$p \in [2,\infty )$$ p ∈ [ 2 , ∞ ) , U is the mild solution, $$U^j$$ U j is obtained from a time discretisation scheme, k is the step size, and $$N_k = T/k$$ N k = T / k for final time $$T>0$$ T > 0 . This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error$$\begin{aligned} \textrm{E}_k {:}{=}\bigg (\sup _{j\in \{0,\ldots ,N_k\}}\mathbb {E}\Vert U(t_j) - U^{j}\Vert _X^p\bigg )^{1/p}, \end{aligned}$$ E k : = ( sup j ∈ { 0 , … , N k } E ‖ U ( t j ) - U j ‖ X p ) 1 / p , which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

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“…The discrete schemes of [23] strongly converge with order 1 − ε in time for arbitrarily small ε > 0. By using an explicit stochastic position Verlet scheme [3], linear convergence was obtained under certain assumptions. Using Itô isometry, authors of [17] constructed a modified stochastic trigonometric method, which achieved superlinear convergence in time.…”

Section: Introductionmentioning

confidence: 99%

Strong approximation for fractional wave equation forced by fractional Brownian motion with Hurst parameter $H\in(0,\frac{1}{2})$

Liu¹

2022

Preprint

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We consider the time discretization of fractional stochastic wave equation with Gaussian noise, which is negatively correlated. Major obstacles to design and analyze time discretization of stochastic wave equation come from the approximation of stochastic convolution with respect to fractional Brownian motion. Firstly, we discuss the smoothing properties of stochastic convolution by using integration by parts and covariance function of fractional Brownian motion. Then the regularity estimates of the mild solution of fractional stochastic wave equation are obtained. Next, we design the time discretization of stochastic convolution by integration by parts. Combining stochastic trigonometric method and approximation of stochastic convolution, the time discretization of stochastic wave equation is achieved. We derive the error estimates of the time discretization. Under certain assumptions, the strong convergence rate of the numerical scheme proposed in this paper can reach 1 2 + H. Finally, the convergence rate and computational efficiency of the numerical scheme are illustrated by numerical experiments.

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Strong Convergence of a Verlet Integrator for the Semilinear Stochastic Wave Equation

Cited by 8 publications

References 30 publications

A symplectic discontinuous Galerkin full discretization for stochastic Maxwell equations

A symplectic discontinuous Galerkin full discretization for stochastic Maxwell equations

Temporal approximation of stochastic evolution equations with irregular nonlinearities

Strong approximation for fractional wave equation forced by fractional Brownian motion with Hurst parameter $H\in(0,\frac{1}{2})$

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