Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

Cohen, David E.; Dujardin, Guillaume

doi:10.1007/s40072-017-0098-1

Cited by 19 publications

(50 citation statements)

References 26 publications

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“…We are now ready to formulate and prove the main result of this section. Recall that u M is the space discrete approximation given by (8) and u M,N is the full discretization given by (22).…”

Section: Resultsmentioning

confidence: 99%

“…It is therefore enough to show that u M,N (t, x) converges to u M (t, x) almost surely, as N → ∞, uniformly in (t, x) and M ∈ N. To achieve this, it suffices to prove that w M,N (t, x) converges to w M (t, x) almost surely in (t, x) as N → ∞. This is because the terms involving u 0 in the approximations u M given by (8) and u M,N given by (22) are the same. We first observe that…”

Section: Full Discretization: Almost Sure Convergencementioning

confidence: 98%

“…8.2]). Finally, we need the following regularity results for the finite difference approximation u M given by (8).…”

Section: Spatial Discretization Of the Stochastic Heat Equationmentioning

confidence: 99%

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A fully discrete approximation of the one-dimensional stochastic heat equation

Anton

Cohen

Quer-Sardanyons

2018

IMA Journal of Numerical Analysis

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A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space-time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz, this explicit time integrator allows for error bounds in L q (Ω), for all q ≥ 2, improving some existing results in the literature. On top of this, we also prove almost sure convergence of the numerical scheme. In the case of non-globally Lipschitz coefficients, we provide sufficient conditions under which the numerical solution converges in probability to the exact solution. Numerical experiments are presented to illustrate the theoretical results.Mathematics Subject Classification (2010): 60H15; 60H35. Error analysis for globally Lipschitz continuous coefficientsThis section is divided into three subsections. We begin by stating the assumptions we will make and by recalling the mild solution of (1). The first subsection is dedicated to

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Section: Resultsmentioning

confidence: 99%

Section: Full Discretization: Almost Sure Convergencementioning

confidence: 98%

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A fully discrete approximation of the one-dimensional stochastic heat equation

Anton

Cohen

Quer-Sardanyons

2018

IMA Journal of Numerical Analysis

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“…Proposition 4.2. The numerical approximation to the linear stochastic Maxwell's equation (11) given by the exponential integrator (12) exactly preserves the following discrete averaged divergence…”

Section: Linear Stochasticmentioning

confidence: 99%

Exponential integrators for stochastic Maxwell's equations driven by Itô noise

Cohen

Cui

Hong

et al. 2020

Journal of Computational Physics

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This article presents explicit exponential integrators for stochastic Maxwell's equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numerical approximation is 1 2 for general multiplicative noise. Combing a proper decomposition with the stochastic Fubini's theorem, the strong order of the proposed scheme is shown to be 1 for additive noise. Moreover, for linear stochastic Maxwell's equation with additive noise, the proposed time integrator is shown to preserve exactly the symplectic structure, the evolution of the energy as well as the evolution of the divergence in the sense of expectation. Several numerical experiments are presented in order to verify our theoretical findings.

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“…Then, we apply the new algorithms to solve the nonlinear Schrödinger equation with highly-oscillatory white noise dispersion (1.3). [8,4,9] are completely innacurate if they do not satisfy the severe timestep restriction h ! ε, we compare the performance of Methods A and B to the performance of the Euler method (3.15).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Multirevolution Integrators for Differential Equations with Fast Stochastic Oscillations

Laurent¹,

Vilmart²

2020

SIAM J. Sci. Comput.

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We introduce a new methodology based on the multirevolution idea for constructing integrators for stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise. Applications include in particular highly-oscillatory Kubo oscillators and spatial discretizations of the nonlinear Schrödinger equation with fast white noise dispersion. We construct a method of weak order two with computational cost and accuracy both independent of the stiffness of the oscillations. A geometric modification that conserves exactly quadratic invariants is also presented.

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Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

Cited by 19 publications

References 26 publications

A fully discrete approximation of the one-dimensional stochastic heat equation

A fully discrete approximation of the one-dimensional stochastic heat equation

Exponential integrators for stochastic Maxwell's equations driven by Itô noise

Multirevolution Integrators for Differential Equations with Fast Stochastic Oscillations

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