Conservative local discontinuous Galerkin methods for a generalized system of strongly coupled nonlinear Schrödinger equations

Mass and energy conservative numerical methods are proposed for a general system of N strongly coupled nonlinear Schrödinger equations (N-CNLS). Motivated by the structure preserving properties of composition methods, two basic conservative, first and second order time integrators, are developed as seed schemes for the derivation of high order conservative methods. To avoid solving a global nonlinear system, involving all the components of the vector field at each time step, a conservative nonlinear splitting method based on a modified Crank-Nicolson scheme is proposed. Conservation of the mass for each component and total energy is formally proved for the semi-discrete primal formulation of the Local Discontinuous Galerkin (LDG) method and for the fully discrete methods. Since the proposed splitting scheme is independent of the spatial discretization, conservation of the same invariants is also obtained for other symmetric discontinuous Galerkin discretizations. Conservation and accuracy of the discrete invariants; and, spatial and temporal convergence are numerically validated on a series of benchmark (2/3)-CNLS systems. Using a special projector operator, the approximated initial energy of the system is shown, numerically, to convergence with order O(h2p+2) when polynomials of degree p are used