A level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials

Abstract

We propose a level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials. The discontinuities in the potential corresponds to potential barriers, at which incoming waves can be partially transmitted and reflected. Previously such a problem was handled by Jin and Wen using the Liouville equation – which arises as the semiclassical limit of the Schrödinger equation – with an interface condition to account for partial transmissions and reflections (S. Jin, X. Wen, SIAM J. Num. Anal. 44 (2006) 1801–1828). However, the initial data are Dirac-delta functions which are difficult to approximate numerically with a high accuracy. In this paper, we extend the level set method introduced in (S. Jin, H. Liu, S. Osher, R. Tsai, J. Comp. Phys. 210 (2005) 497–518) for this problem. Instead of directly discretizing the Delta functions, our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow, and hence can be solved much more easily using conventional high order discretization schemes. Two ideas are introduced here: (1) The solutions of the problems involving partial transmissions and partial reflections are decomposed into a finite sum of solutions solving problems involving only complete transmissions and those involving only complete reflections. For problems involving only complete transmission or complete reflection, the method of JLOT applies and is used in our simulations; (2) A reinitialization technique is introduced so that waves coming from multiple transmissions and reflections can be combined seamlessly as new initial value problems. This is implemented by rewriting the sum of several delta functions as one delta function with a suitable weight, which can be easily implemented numerically. We carry out numerical experiments in both one and two space dimensions to verify this new algorithm.