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Journal of Computational Mathematics ›› 2018, Vol. 36 ›› Issue (2): 276-309.DOI: 10.4208/jcm.1701-m2016-0525

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EXPONENTIAL INTEGRATORS FOR STOCHASTIC SCHRÖDINGER EQUATIONS DRIVEN BY ITÔ NOISE

Rikard Anton, David Cohen   

  1. Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden Department of Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria
  • Received:2016-01-26 Revised:2016-11-08 Online:2018-03-15 Published:2018-03-15
  • Supported by:

    This work was partially supported by UMIT Research Lab at Umeå University and the Swedish Research Council (VR)(project nr.2013 -4562).

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Rikard Anton, David Cohen. EXPONENTIAL INTEGRATORS FOR STOCHASTIC SCHRÖDINGER EQUATIONS DRIVEN BY ITÔ NOISE[J]. Journal of Computational Mathematics, 2018, 36(2): 276-309.

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.
Key words: Stochastic partial differential equations, Stochastic Schrö, dinger equations, Numerical methods, Geometric numerical integration, Stochastic exponential integrators, Strong convergence, Trace formulas

CLC Number: 

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