Volume 7, Issue 6
Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

Emil Kieri, Gunilla Kreiss & Olof Runborg

Adv. Appl. Math. Mech., 7 (2015), pp. 687-714.

Published online: 2018-05

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In the semiclassical regime, solutions to the time-dependent Schrödinger  equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.

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@Article{AAMM-7-687, author = {}, title = {Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {7}, number = {6}, pages = {687--714}, abstract = {

In the semiclassical regime, solutions to the time-dependent Schrödinger  equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2013.m411}, url = {http://global-sci.org/intro/article_detail/aamm/12071.html} }
TY - JOUR T1 - Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 687 EP - 714 PY - 2018 DA - 2018/05 SN - 7 DO - http://doi.org/10.4208/aamm.2013.m411 UR - https://global-sci.org/intro/article_detail/aamm/12071.html KW - AB -

In the semiclassical regime, solutions to the time-dependent Schrödinger  equation for molecular dynamics are highly oscillatory. The number of grid points required for resolving the oscillations may become very large even for simple model problems, making solution on a grid intractable. Asymptotic methods like Gaussian beams can resolve the oscillations with little effort and yield good approximations when the atomic nuclei are heavy and the potential is smooth. However, when the potential has variations on a small length-scale, quantum phenomena become important. Then asymptotic methods are less accurate. The two classes of methods perform well in different parameter regimes. This opens for hybrid methods, using Gaussian beams where we can and finite differences where we have to. We propose a new method for treating the coupling between the finite difference method and Gaussian beams. The new method reduces the needed amount of overlap regions considerably compared to previous methods, which improves the efficiency.

Emil Kieri, Gunilla Kreiss & Olof Runborg. (2020). Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations. Advances in Applied Mathematics and Mechanics. 7 (6). 687-714. doi:10.4208/aamm.2013.m411
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