Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrödinger equations

Using a unified framework, the formulation of a super-convergent discontinuous Galerkin (SDG) method and a hybridized discontinuous Galerkin (HDG) version, both applied to a general nonlinear Schrödinger equation is presented. Conservation of the mass and the energy is studied, theoretically for the semi-discrete formulation; and, for the fully discrete method using the Modified Crank–Nicolson time scheme. Conservation of both quantities is numerically validated on two dimensional problems and high order approximations. A numerical study of convergence illustrates the advantages of the new formulations over the traditional Local Discontinuous Galerkin (LDG) method. Numerical experiments show that the approximation of the initial discrete energy converges with order 2k+1, which is better than that obtained by the standard (continuous) finite element, which is only of order 2k when polynomials of degree k are used