Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
A PDE-based fast local level set method
Journal of Computational Physics
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Journal of Computational Physics
High-frequency wave propagation by the segment projection method
Journal of Computational Physics
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
Journal of Computational Physics
A level set based Eulerian method for paraxial multivalued traveltimes
Journal of Computational Physics
Local level set method in high dimension and codimension
Journal of Computational Physics
Journal of Computational Physics
A Local Level Set Method for Paraxial Geometrical Optics
SIAM Journal on Scientific Computing
Journal of Computational Physics
A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations
Journal of Scientific Computing
Computing multi-valued velocity and electric fields for 1D Euler--Poisson equations
Applied Numerical Mathematics
SIAM Journal on Scientific Computing
Journal of Computational Physics
Superposition of Multi-Valued Solutions in High Frequency Wave Dynamics
Journal of Scientific Computing
Journal of Scientific Computing
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials
Journal of Computational Physics
Hi-index | 31.45 |
A novel Bloch band based level set method is proposed for computing the semiclassical limit of Schrodinger equations in periodic media. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the WKB approximation and homogenization leads to the Bloch eigenvalue problem and an associated Hamilton-Jacobi system for the phase in each Bloch band, with the Bloch eigenvalue be part of the Hamiltonian. We formulate a level set description to capture multi-valued solutions to the band WKB system, and then evaluate total homogenized density over a sample set of bands. A superposition of band densities is established over all bands and solution branches when away from caustic points. The numerical approach splits the solution process into several parts: (i) initialize the level set function from the band decomposition of the initial data; (ii) solve the Bloch eigenvalue problem to compute Bloch waves; (iii) evolve the band level set equation to compute multi-valued velocity and density on each Bloch band; (iv) evaluate the total position density over a sample set of bands using Bloch waves and band densities obtained in steps (ii) and (iii), respectively. Numerical examples with different number of bands are shown to demonstrate the capacity of the method.