A Parallel Time-Propagation Solver for the Non-Linear Schroedinger Equation.

We describe a powerful numerical method for solving the time-dependent non-linear Schr\"{o}dinger equation. Our method is based on the finite-element discrete variable representation. The time-propagation is facilitated either by the Lanczos-Arnoldi method or by split-operator formulas of different orders. The ground-state solution is found by propagation in imaginary time using an adaptive time stepping algorithm, and the absolute convergence of the propagation is faithfully characterized by a positive-definite error norm. Parallelization of this method is transparent, and we have utilized an MPI implementation demonstrating linear scaling of wall-clock computation time with the number of processors used.