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Journal of Scientific Computing

, Volume 77, Issue 3, pp 1444–1467 | Cite as

On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method

  • P. Castillo
  • S. Gómez
Article
  • 120 Downloads

Abstract

Using the primal formulation of the Local Discontinuous Galerkin (LDG) method, discrete analogues of the energy and the Hamiltonian of a general class of fractional nonlinear Schrödinger equation are shown to be conserved for two stabilized version of the method. Accuracy of these invariants is numerically studied with respect to the stabilization parameter and two different projection operators applied to the initial conditions. The fully discrete problem is analyzed for two implicit time step schemes: the midpoint and the modified Crank–Nicolson; and the explicit circularly exact Leapfrog scheme. Stability conditions for the Leapfrog scheme and a stabilized version of the LDG method applied to the fractional linear Schrödinger equation are derived using a von Neumann stability analysis. A series of numerical experiments with different nonlinear potentials are presented.

Keywords

Fractional nonlinear Schrödinger equation (FNLS) Local discontinuous Galerkin (LDG) Energy and Hamiltonian conservation CFL 

Mathematics Subject Classification

65M12 65M20 65M60 

Notes

Acknowledgements

We kindly thank the anonymous reviewers for their valuable suggestions.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Puerto RicoMayagüezPuerto Rico

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