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

Eulerian Gaussian beam method for high frequency wave propagation in heterogeneous media with discontinuities in one direction

“…We correct the flux at the interface by proper reflection and transmission conditions. This leads to an important reformulation of the level set functions (2.6): when the wave reaches the interface, the level sets given in (2.6) become discontinuous and hard to be computed due to the interaction of waves ( [26]), and it turns out we need more than one set of level set functions to describe the reflected and transmitted beams.…”

“…The contribution of these functions to the final Gaussian beam solution will be described by the level set functions φ r,tr ± . This is essentially the same idea as introduced in [26] for computing the semiclassical limit of the Schrödinger equation with discontinuous potentials. The details of the formulations for S, A, and φ are given as below.…”

“…If the initial wave comes from both sides of the interface, we can simply split it into left-coming and right-coming parts and apply the algorithm twice. For the case of multiple interfaces where a wave packet hits interfaces multiple times, we will need the method proposed in [26]. Briefly speaking, if the interfaces divide R n into K disjoint parts R n = K j=1 Ω j , one can decompose the original problem into K + 1 systems, and the Gaussian beam solution takes the following form:…”

On the Quasi-Random Choice Method for the Liouville Equation of Geometrical Optics with Discontinuous Wave Speed

“…The discontinuity corresponds to an interface, at which incoming waves can be partially transmitted and reflected. Against this background, much work has been done in the past both analytically [9][10][11][12] and numerically [13][14][15][16][17][18]. Numerically this problem consists of three challenges:…”

“…However, as pointed out in [8,14], it only readily [18] suggested a way to handle this problem by the level set method, but it needs an initialization procedure at every time step. The random choice method (RCM), or the Glimm's scheme, was first introduced by Glimm in 1965 for proving the existence of global weak solutions to hyperbolic system of conservation laws [21].…”