Energy conserving discontinuous Galerkin method with scalar auxiliary variable technique for the nonlinear Dirac equation

Abstract

In this paper, we propose a fully-discrete energy-conserving scheme for the nonlinear Dirac equation, by combining the scalar auxiliary variable (SAV) technique with discontinuous Galerkin (DG) discretization. We start by discussing the semi-discrete DG discretization, and show that, with suitable choices of numerical fluxes, the resulting method conserves the charge, energy exactly and preserves the multi-symplectic structure. The optimal error estimate of semi-discrete DG scheme is carried out. We combine it with the energy conserving SAV technique, and demonstrate that the fully-discrete scheme conserves the discrete global energy exactly. Both second order SAV method based on the midpoint rule and its high order extension have been studied. The proposed methods have been tested on some numerical experiments, which confirm the optimal rates of convergence and the energy conserving property. Numerical comparison with energy dissipative DG method is also provided to demonstrate that the numerical error of energy conserving method does not grow significantly in long time simulations.