Galerkin/Runge-Kutta discretization for the nonlinear Schrödinger equation

A class of fully discrete high order Galerkin Runge-Kutta methods are constructed and analyzed for the nonlinear Schrodinger equation. Optimal order error estimates are established for the 0-boundary and periodic boundary value problems, and several computational results such as the order of the temporal accuracy, preservation of two invariants, various kinds of errors are presented. Furthermore, it is noted that these methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.