An energy-based discontinuous Galerkin method for the nonlinear Schrödinger equation with wave operator

This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schr\"odinger equation with the wave operator. The focus of the study is on the energy-conserving or energy-dissipating behavior of the method with some simple mesh-independent numerical fluxes we designed. We establish error estimates in the energy norm that require careful selection of a test function for the auxiliary equation involving the time derivative of the displacement variable. A critical part of the convergence analysis is to establish the L2 error bounds for the time derivative of the approximation error in the displacement variable by using the equation that determines its mean value. Using a specially chosen test function, we show that one can create a linear system for the time evolution of the unknowns even when dealing with nonlinear properties in the original problem. Extensive numerical experiments are provided to demonstrate the optimal convergence of the scheme in the L2 norm with our choices of the numerical flux.