Energy preserving model order reduction of the nonlinear Schrödinger equation

  • Bülent Karasözen
  • Murat Uzunca


An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.


Nonlinear Schrödinger equation Discontinuous Galerkin method Average vector field method Proper orthogonal decomposition Discrete empirical interpolation method Dynamic mode decomposition 

Mathematics Subject Classification (2010)

65P10 65M60 365Q55 37M15 93A15 


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The authors would like to thank the reviewers for the comments and suggestions that helped to improve the manuscript.


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Authors and Affiliations

  1. 1.Institute of Applied Mathematics & Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of MathematicsSinop UniversitySinopTurkey

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