Multi-phase computations in geometrical optics
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
Sticky Particles and Scalar Conservation Laws
SIAM Journal on Numerical Analysis
Semi-Lagrangian methods for level set equations
Journal of Computational Physics
Fast tree-based redistancing for level set computations
Journal of Computational Physics
A PDE-based fast local level set method
Journal of Computational Physics
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Journal of Computational Physics
Using K-branch entropy solutions for multivalued geometric optics computations
Journal of Computational Physics
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations
Mathematics of Computation
Regularization Techniques for Numerical Approximation of PDEs with Singularities
Journal of Scientific Computing
Simplicial isosurfacing in arbitrary dimension and codimension
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Computing multi-valued velocity and electric fields for 1D Euler--Poisson equations
Applied Numerical Mathematics
Journal of Computational Physics
High order numerical methods to a type of delta function integrals
Journal of Computational Physics
Superposition of Multi-Valued Solutions in High Frequency Wave Dynamics
Journal of Scientific Computing
Journal of Scientific Computing
Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime
Journal of Computational Physics
A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations
Journal of Computational Physics
Discretizing delta functions via finite differences and gradient normalization
Journal of Computational Physics
High order numerical methods to two dimensional delta function integrals in level set methods
Journal of Computational Physics
Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials
Journal of Computational Physics
Journal of Computational Physics
A level set approach for dilute non-collisional fluid-particle flows
Journal of Computational Physics
High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods
SIAM Journal on Scientific Computing
Realizable high-order finite-volume schemes for quadrature-based moment methods
Journal of Computational Physics
Conditional quadrature method of moments for kinetic equations
Journal of Computational Physics
A Hybrid Phase Flow Method for Solving the Liouville Equation in a Bounded Domain
SIAM Journal on Numerical Analysis
Strong Stability Preserving Two-step Runge-Kutta Methods
SIAM Journal on Numerical Analysis
Hi-index | 31.53 |
We develop a level set method for the computation of multivalued physical observables (density, velocity, etc.) for the semiclassical limit of the Schrodinger equation. This method uses an Eulerian formulation and applies directly to arbitrary number of space dimensions. The main idea is to evolve the density near an n-dimensional manifold that is identified as the common zeros of n level set functions in phase space. These level set functions are generated from solving the Liouville equation with initial data chosen to embed the phase gradient. Simultaneously we track a new quantity f by solving again the Liouville equation near the obtained zero level set but with initial density as initial data. The multivalued density and higher moments are thus resolved by integrating f along the n-dimensional manifold in the phase directions. We show that this is equivalent to using the Wigner approach but decomposing the velocity from the density, each of which evolves by the same Liouville equation. The main advantages of this approach, in contrast to the standard kinetic equation approach using the Liouville equation with a Dirac measure initial data, include: (1) the Liouville equations are solved with L^~ initial data, and a singular integral involving the Dirac-@d function is evaluated only in the post-processing step, thus avoiding oscillations and excessive numerical smearing; (2) a local level set method can be utilized to significantly reduce the computation in the phase space. These advantages allow us to compute, for the first time, all physical observables for multidimensional problems in an Eulerian framework.