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APS

Wavepacket Dynamics on Dynamically Adapting Grids: Application of the Equidistribution Principle

Keith H. Hughes, Róbert E. Wyatt

Published Mar 1, 2003• 0 citations
OpenAlex

Abstract

A moving grid approach to wavepacket dynamics is described that enables grid points to be used efficiently in regions where high resolution of the wavepacket is required. The grid movement is based on the principle of equidistribution and by using a grid smoothing technique the grid points trace a path that continuously adapt to reflect the dynamics of the wavepacket. The technique is robust and allows accurate computations to be obtained for long wavepacket propagation times. Results are presented for two systems: tunnelling dynamics in a double well potential and scattering of a wavepacket from a repulsive Eckart barrier.

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A moving grid approach to wavepacket dynamics is described that enables grid points to be used efficiently in regions where high resolution of the wavepacket is required. The grid movement is based on the principle of equidistribution and by using a grid smoothing technique the grid points trace a path that continuously adapt to reflect the dynamics of the wavepacket. The technique is robust and allows accurate computations to be obtained for long wavepacket propagation times. Results are presented for two systems: tunnelling dynamics in a double well potential and scattering of a wavepacket from a repulsive Eckart barrier.
A moving grid approach to a dynamical study of dissipative systems is described. The dynamics are studied in phase space for the Caldeira-Leggett master equation. The grid movement is based on the principle of equidistribution and, by using a grid smoothing technique, the grid points trace a path that continuously adapts to reflect the dynamics of a phase-space distribution function. The technique is robust and allows accurate computations to be obtained for long propagation times. The effects of dissipation on the dynamics are studied and results are presented for systems subject to both periodic and nonperiodic multiminimum potential functions.
The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.
Abstract The moving boundary truncated grid (TG) method is used to study wave packet dynamics of multidimensional quantum systems. As time evolves, appropriate Eulerian grid points required for propagating a wave packet are activated and deactivated with no advance information about the dynamics. This method is applied to the Henon‐Heiles potential and wave packet barrier scattering in two, three, and four dimensions. Computational results demonstrate that the TG method not only leads to a great reduction in the number of grid points needed to perform accurate calculations but also is computationally more efficient than the full grid calculations.
A new hybrid moving grid approach to wave packet dynamics is described. Exterior points within a nonrigid frame follow Lagrangian trajectories obtained by solving the hydrodynamic equations of motion. Internal grid points within one or more open windows follow non-Lagrangian adaptive paths. Within these windows, problems encountered with quantum trajectories near wave function nodes are circumvented by directly solving the moving path transform of the Schrödinger equation. Excellent results are obtained for evolution of the density in a double well potential even though multiple ripples develop in the density.
Time-dependent wavepackets are widely used to model various phenomena in physics. One approach in simulating the wavepacket dynamics is the quantum trajectory method (QTM). Based on the hydrodynamic formulation of quantum mechanics, the QTM represents the wavepacket by an unstructured set of pseudoparticles whose trajectories are coupled by the quantum potential. The governing equations for the pseudoparticle trajectories are solved using a computationally intensive moving weighted least squares (MWLS) algorithm, and the trajectories can be computed in parallel. This paper contributes a strategy for improving the performance of wavepacket simulations using the QTM. Specifically, adaptivity is incorporated into the MWLS algorithm, and loop scheduling techniques are employed to dynamically load balance the parallel computation of the trajectories. The adaptive MWLS algorithm reduces the amount of computations without sacrificing accuracy, while adaptive loop scheduling addresses the load imbalance introduced by the algorithm and the runtime system. Results of experiments on a Linux cluster are presented to confirm that the adaptive MWLS reduces the trajectory computation time by up to 24%, and adaptive loop scheduling achieves parallel efficiencies of up to 85% when simulating a free particle.