# Energy preserving model order reduction of the nonlinear Schr\"odinger equation

## TL;DR

This paper develops an energy-preserving reduced order model for the 2D nonlinear Schrödinger equation, ensuring long-term stability and efficient computation using POD-Galerkin, DEIM, and DMD methods.

## Contribution

It introduces a novel energy-preserving ROM for the 2D NLSE that combines POD-Galerkin with DEIM and DMD for efficient nonlinear computations.

## Key findings

- The ROM preserves energy and mass, ensuring stability.
- POD-DMD significantly speeds up computations compared to POD-DEIM.
- Both methods accurately approximate the full order model.

## Abstract

An energy preserving reduced order model is developed for two dimensional nonlinear Schr\"odinger 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.