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

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.

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Acknowledgements

We kindly thank the anonymous reviewers for their valuable suggestions.

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Correspondence to P. Castillo.

Additional information

Dedicated to Prof. Bernardo Cockburn and Prof. Chi-Wang Shu.

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Castillo, P., Gómez, S. On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method. J Sci Comput 77, 1444–1467 (2018). https://doi.org/10.1007/s10915-018-0708-8

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Keywords

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

Mathematics Subject Classification

  • 65M12
  • 65M20
  • 65M60