Volume 36, Issue 2
Exponential Integrators for Stochastic Schrödinger Equations Driven by Itô Noise

Rikard AntonDavid Cohen

J. Comp. Math., 36 (2018), pp. 276-309.

Published online: 2018-04

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  • Abstract

We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schrödinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

  • Keywords

Stochastic partial differential equations, Stochastic Schrödinger equations, Numerical methods, Geometric numerical integration, Stochastic exponential integrators, Strong convergence, Trace formulas.

  • AMS Subject Headings

35Q55, 60H15, 65C20, 65C30, 65C50, 65J08

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

rikard.anton@umu.se (Rikard Anton)

david.cohen@umu.se (David Cohen)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-276, author = {Anton , Rikard and Cohen , David}, title = {Exponential Integrators for Stochastic Schrödinger Equations Driven by Itô Noise}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {2}, pages = {276--309}, abstract = {

We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schrödinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1701-m2016-0525}, url = {http://global-sci.org/intro/article_detail/jcm/12259.html} }
TY - JOUR T1 - Exponential Integrators for Stochastic Schrödinger Equations Driven by Itô Noise AU - Anton , Rikard AU - Cohen , David JO - Journal of Computational Mathematics VL - 2 SP - 276 EP - 309 PY - 2018 DA - 2018/04 SN - 36 DO - http://doi.org/10.4208/jcm.1701-m2016-0525 UR - https://global-sci.org/intro/article_detail/jcm/12259.html KW - Stochastic partial differential equations, Stochastic Schrödinger equations, Numerical methods, Geometric numerical integration, Stochastic exponential integrators, Strong convergence, Trace formulas. AB -

We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schrödinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

Rikard Anton & David Cohen. (2020). Exponential Integrators for Stochastic Schrödinger Equations Driven by Itô Noise. Journal of Computational Mathematics. 36 (2). 276-309. doi:10.4208/jcm.1701-m2016-0525
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