Fast Computational Methods For High Frequency Waves

The numerical computation of high frequency waves is an important and popu-lar topics in applied maths since they arise in many applications, including quantummechanics, acoustic waves, optics waves, elastic waves and electromagnetic waves.The major challenge in these problems is that the direct numerical simulation is pro-hibitively expensive when the wave length is extremely small compared to the overallsize of the computational domain. Therefore, approximation method based on asymp-totic analysis are needed. In this thesis, we develop new Geometrical optics approx-imation and Gaussian beam method which are e?ective computational tools for highfrequency waves problems.As the first part of our work, we consider the Geometrical optics approximationbased computational model to the high frequency wave propagation in heterogeneousmedia. In general, waves would be re?ected and transmitted when crossing the discon-tinuous potential or medium interface. This leads to poor or even incorrect numericalresolution for standard method of the Geometrical optics approximation without spe-cial considerations. S. Jin and X. Wen proposed the Hamiltonian-preserving schemes tohandle this di?culty. In this thesis, we combine this idea with the phase-?ow method,and then introduce a fast algorithm, which is called hybrid phase-?ow method, for com-puting high frequency waves propagation in heterogeneous media based Geometricaloptics approximation. The numerical results demonstrate the e?ciency and accuracyof this method.A problem with Geometrical optics approximation is that the asymptotic solutionis invalid at caustics. The Gaussian beam method is an e?cient approximation methodthat allows accurate computation of the wave around caustics. In the second part ofthis thesis, we solve the Schro¨dinger equation using both Lagrangian and Eulerianformulations of the Gaussian beam methods. A new Eulerian Gaussian beam methodusing the level set method based only on solving a few Liouville equations is developed. The major innovation here is that we can construct the Hessian matrices of the beams bytaking derivatives of the existing level set function. This greatly improves the e?ciencyfor the Eulerian Gaussian beam method. Comparing to the Eulerian formulation of thegeometrical optics, we gain better accuracy around caustics while the computationalcomplexity is almost the same. Through several numerical experiments, we verifyour conclusions. In addition, we numerically study the selection of parameters in theGaussian beam method. As an extension of the Gaussian beam method, we proposethe bloch decomposition-based Gaussian beam method for the Schro¨dinger equationwith periodic potentials, and numerically study the Gaussian beam methods for one-dimensional nonlinear Schro¨dinger-Poisson equations.