Conservative local discontinuous Galerkin methods for the cubic-quintic nonlinear Schrödinger equation

In this paper, we propose the local discontinuous Galerkin methods to solve the cubic-quintic nonlinear Schrödinger equation. Our numerical methods are based on the local discontinuous Galerkin methods in space and Crank-Nicholson method in time. By choosing the appropriate numerical fluxes, we can prove the mass- and energy-conserving properties for both the semi-discrete and fully discrete methods. We also present some numerical experiments, including soliton with linear potential and with time sinusoidal modulated potential, to demonstrate the accuracy and mass- and energy-conserving properties of our proposed methods.