An Energy-Based Discontinuous Galerkin Method for the Nonlinear Schrödinger Equation with Wave Operator

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2459-2483, December 2024.
Abstract. This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schrödinger 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 weak formulation for the auxiliary equation involving the time derivative of the displacement variable. A critical part of the convergence analysis is to establish the [math] 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 special weak formulation, 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. Numerical experiments were performed to demonstrate the optimal convergence of the scheme in the [math] norm. These experiments involved specific choices of numerical fluxes combined with specific choices of approximation spaces.