Mass preserving discontinuous Galerkin methods for Schrödinger equations

We construct, analyze and numerically validate a class of mass preserving, direct discontinuous Galerkin (DDG) schemes for Schrödinger equations subject to both linear and nonlinear potentials. Up to round-off error, these schemes preserve the discrete version of the mass of the continuous solution. For time discretization, we use the Crank-Nicolson for linear Schrödinger equations, and the Strang splitting for nonlinear Schrödinger equations, so that numerical mass is still preserved at each time step. The DDG method when applied to linear Schrödinger equations is shown to have the optimal (k+1)th order of accuracy for polynomial elements of degree k. The numerical tests demonstrate both accuracy and capacity of these methods.