Parallel solver for the time-dependent linear and nonlinear Schrödinger equation

A solution of the time-dependent Schrödinger equation is required in a variety of problems in physics and chemistry. These include atoms and molecules in time-dependent electromagnetic fields, time-dependent approaches to atomic collision problems, and describing the behavior of materials subjected to internal and external forces. We describe an approach in which the finite-element discrete variable representation (FEDVR) is combined with the real-space product (RSP) algorithm to generate an efficient and highly accurate method for the solution of the time-dependent linear and nonlinear Schrödinger equation. The FEDVR provides a highly accurate spatial representation using a minimum number of grid points (N) while the RSP algorithm propagates the wave function in O(N) operations per time step. Parallelization of the method is transparent and is implemented here by distributing one or two spatial dimensions across the available processors, within the message-passing-interface scheme. The complete formalism and a number of three-dimensional examples are given; its high accuracy and efficacy are illustrated by a comparison with the usual finite-difference method.