Mass preserving discontinuous Galerkin methods for Schrodinger equations

We construct, analyze and numerically validate a class of mass preserving, direct discontinuous Galerkin (DDG) schemes for Schrodinger 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 Schrodinger equations, and the Strang splitting for nonlinear Schrodinger equations, so that numerical mass is still preserved at each time step. The DDG method when applied to linear Schrodinger 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. (C) 2014 Elsevier Inc. All rights reserved.