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Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations

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Title: Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations
Authors: Bréhier, Charles-Edouard, Cohen, David, 1977
Source: Numerisk analys och simulering av PDE med slumpmässig dispersion Applied Numerical Mathematics. 186:57-83
Subject Terms: Geometric numerical integration, Stochastic partial differential equations, Strong convergence, Trace formulas, Stochastic Schrödinger equations, Splitting integrators
Description: We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schrödinger equations driven by additive Itô noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
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Access URL: https://research.chalmers.se/publication/534303
https://research.chalmers.se/publication/527569
https://arxiv.org/abs/2007.02354
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Items – Name: Title
  Label: Title
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  Data: Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations
– Name: Author
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  Data: <searchLink fieldCode="AR" term="%22Bréhier%2C+Charles-Edouard%22">Bréhier, Charles-Edouard</searchLink><br /><searchLink fieldCode="AR" term="%22Cohen%2C+David%22">Cohen, David</searchLink>, 1977
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  Data: <i>Numerisk analys och simulering av PDE med slumpmässig dispersion Applied Numerical Mathematics</i>. 186:57-83
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  Data: <searchLink fieldCode="DE" term="%22Geometric+numerical+integration%22">Geometric numerical integration</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+partial+differential+equations%22">Stochastic partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Strong+convergence%22">Strong convergence</searchLink><br /><searchLink fieldCode="DE" term="%22Trace+formulas%22">Trace formulas</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+Schrödinger+equations%22">Stochastic Schrödinger equations</searchLink><br /><searchLink fieldCode="DE" term="%22Splitting+integrators%22">Splitting integrators</searchLink>
– Name: Abstract
  Label: Description
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  Data: We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schrödinger equations driven by additive Itô noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
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        Value: 10.1016/j.apnum.2023.01.002
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      – Text: English
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        PageCount: 27
        StartPage: 57
    Subjects:
      – SubjectFull: Geometric numerical integration
        Type: general
      – SubjectFull: Stochastic partial differential equations
        Type: general
      – SubjectFull: Strong convergence
        Type: general
      – SubjectFull: Trace formulas
        Type: general
      – SubjectFull: Stochastic Schrödinger equations
        Type: general
      – SubjectFull: Splitting integrators
        Type: general
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      – TitleFull: Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations
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            NameFull: Bréhier, Charles-Edouard
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            NameFull: Cohen, David
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              Value: 186
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            – TitleFull: Numerisk analys och simulering av PDE med slumpmässig dispersion Applied Numerical Mathematics
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