Advances in Applied Mathematics and Mechanics

August 24, 2020

Journal of Advances in

Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

a1 Division of Scientific Computing, Department of Information Technology, Uppsala University, Sweden

a2 Department of Mathematics and Swedish e-Science Research Center (SeRC), KTH, Sweden


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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