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 $\left(N\right)$ while the RSP algorithm propagates the wave function in $O\left(N\right)$ 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.

Figure 1

(Color online) The scaling test of RSP-FEDVR on a supercomputer “QSC” of LANL for the ground state of Bose-Einstein condensate in a 3D cylindrical trap (case 1), which is spanned by a grid size of $4097\times 81\times 81\sim 2.7\times {10}^7$ points. Since such an elongated data set favors our 1D-decomposition scheme, we are observing an “superscaling” from 4 to 128 CPU’s.

Figure 2

(Color online) The scaling test of RSP-FEDVR for a larger and less elongated data. We treat the same Bose-Einstein condensate as in Fig. 1 but on a larger 3D grid, which is spanned by a grid size of $2049\times 513\times 513\sim 5.4\times {10}^8$ points. We show both the “1D-decomposition” and “2D-decomposition” results, respectively, in (blue) diamonds and (red) squares. The “1D decomposition” begins to break down as the number of CPU’s increases from $N=128$ to $N=256$. However, it shows excellent linear scaling up to $n=256$ CPU’s in the case of “2D decomposition.”

Figure 3

(Color online) Other examples of the “2D-decomposition” parallelization with the RSP-FEDVR code. Same BEC test model (case 1) as in previous two figures. Tests are performed for different data sets (marked in the figure) on the two LANL’s supercomputers “Flash” and “QSC,” respectively.

Figure 4

(Color online) Tests of the RSP-FEDVR scaling on the grid size of the 3D BEC problem. It shows a perfect linear scaling over grid sizes varying by four orders of magnitude.

Figure 5

The total energy versus time during the imaginary-time relaxation of a trial wave packet to the ground state of the helium atom.

Figure 6

(Color online) The ground-state probability of He on the $(r_1,r_2)$ plane.

Figure 7

(Color online) The snapshots of electron probability of a hydrogen atom, exposed to a CP-FCP, are shown for instants of (a) $t=3.0\mathrm{fs}$ and (b) $t=5.0\mathrm{fs}$. The laser pulse has a duration of $5\mathrm{fs}$, a wavelength of $800\mathrm{nm}$, and an intensity of $2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The carrier-envelope phase is equal to 60°, and the laser field is left-hand circularly polarized.