Exponential Integrators for Stochastic Schrödinger Equations Driven by Itô Noise

Authors

  • Rikard Anton Department of Mathematics and Mathematical Statistics, Ume˚a University, SE–901 87 Ume˚a, Sweden
  • David Cohen Department of Mathematics and Mathematical Statistics, Ume˚a University, SE–901 87 Ume˚a, Sweden

DOI:

https://doi.org/10.4208/jcm.1701-m2016-0525

Keywords:

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

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.

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Published

2019-02-12

Issue

Vol. 36 No. 2 (2018)

Section

Articles