Exponential integrators for nonlinear schrödinger equations with white noise dispersion

David Cohen; Guillaume Dujardin

doi:10.1007/s40072-017-0098-1

Exponential integrators for nonlinear schrödinger equations with white noise dispersion

David Cohen
, Guillaume Dujardin

École Polytechnique

Umeå University
University of Innsbruck
CNRS UMR 8524

Research output: Contribution to journal › Article › peer-review

18 Citations (Scopus)

Abstract

This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L²-norm of the solution.

Original language	English
Pages (from-to)	592-613
Number of pages	22
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Volume	5
Issue number	4
DOIs	https://doi.org/10.1007/s40072-017-0098-1
Publication status	Published - 1 Dec 2017

Keywords

Exponential integrators
Geometric numerical integration
Mean-square convergence
Nonlinear Schrödinger equation
Numerical methods
Stochastic partial differential equations
White noise dispersion

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10.1007/s40072-017-0098-1

Cite this

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title = "Exponential integrators for nonlinear schr{\"o}dinger equations with white noise dispersion",

abstract = "This article deals with the numerical integration in time of the nonlinear Schr{\"o}dinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L2-norm of the solution.",

keywords = "Exponential integrators, Geometric numerical integration, Mean-square convergence, Nonlinear Schr{\"o}dinger equation, Numerical methods, Stochastic partial differential equations, White noise dispersion",

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AU - Cohen, David

AU - Dujardin, Guillaume

N1 - Publisher Copyright: © The Author(s) 2017. This article is an open access publication.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L2-norm of the solution.

AB - This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L2-norm of the solution.

KW - Exponential integrators

KW - Geometric numerical integration

KW - Mean-square convergence

KW - Nonlinear Schrödinger equation

KW - Numerical methods

KW - Stochastic partial differential equations

KW - White noise dispersion

UR - https://www.scopus.com/pages/publications/85055971953

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DO - 10.1007/s40072-017-0098-1

M3 - Article

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VL - 5

SP - 592

EP - 613

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

IS - 4

ER -

Exponential integrators for nonlinear schrödinger equations with white noise dispersion

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Keywords

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