Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Discrete Gabor transforms with complexity O(NlogN)
Signal Processing
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Journal of Computational Physics
A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schrödinger Equation
SIAM Journal on Numerical Analysis
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
Radial Basis Functions
Simplicial isosurfacing in arbitrary dimension and codimension
Journal of Computational Physics
A level set based Eulerian method for paraxial multivalued traveltimes
Journal of Computational Physics
A Local Level Set Method for Paraxial Geometrical Optics
SIAM Journal on Scientific Computing
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation
Journal of Computational Physics
The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation
Journal of Computational Physics
High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods
SIAM Journal on Scientific Computing
An Eulerian approach for computing the finite time Lyapunov exponent
Journal of Computational Physics
Fast multiscale Gaussian beam methods for wave equations in bounded convex domains
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.48 |
We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrodinger equation. Traditional Gaussian beam type methods for the Schrodinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61-SM76], we develop a new Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrodinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi-classical solutions to the Schrodinger equation. For instance, in the one-dimensional case the proposed algorithm requires only O(sNm2) operations to compute s different solutions with s different initial wave functions under the influence of the same potential, where N=O(1/), is the Planck constant, and m N is the number of computed beams which depends on weakly. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semi-classical solutions even at caustics.