Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime (Communications in Computational Physics, 2011)
A novel Eulerian Gaussian beam method was developed in [8] to computethe Schrödinger equation efficiently in the semiclassical regime. In this paper, we introducean efficient semi-Eulerian implementation of this method. The new algorithminherits the essence of the Eulerian Gaussian beam method where the Hessian is computedthrough the derivatives of the complexified level set functions instead of solvingthe dynamic ray tracing equation. The difference lies in that, we solve the ray tracingequations to determine the centers of the beams and then compute quantities of interestsonly around these centers. This yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of thephase plane. As a consequence, it reduces the computational cost and also avoids thedelicate issue of beam summation around the caustics in the Eulerian Gaussian beammethod. Moreover, the semi-Eulerian Gaussian beam method can be easily generalizedto higher order Gaussian beam methods, which is the topic of the second partof this paper. Several numerical examples are provided to verify the accuracy andefficiency of both the first order and higher order semi-Eulerian methods.