## Energy preserving model order reduction of the nonlinear Schrodinger equation

ADVANCES IN COMPUTATIONAL MATHEMATICS, vol.44, no.6, pp.1769-1796, 2018 (Journal Indexed in SCI)  • Publication Type: Article / Article
• Volume: 44 Issue: 6
• Publication Date: 2018
• Doi Number: 10.1007/s10444-018-9593-9
• Title of Journal : ADVANCES IN COMPUTATIONAL MATHEMATICS
• Page Numbers: pp.1769-1796
• Keywords: Nonlinear Schrodinger equation, Discontinuous Galerkin method, Average vector field method, Proper orthogonal decomposition, Discrete empirical interpolation method, Dynamic mode decomposition, DISCONTINUOUS GALERKIN METHODS, PROPER ORTHOGONAL DECOMPOSITION, EMPIRICAL INTERPOLATION METHOD, MULTI-SYMPLECTIC FORMULATIONS, MISSING POINT ESTIMATION, HAMILTONIAN-SYSTEMS, MATLAB TOOLBOX, DYNAMICS, CONSERVATION, INTEGRATORS

#### Abstract

An energy preserving reduced order model is developed for two dimensional nonlinear Schrodinger 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.