Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
The sharpness of Kuznetsov's ODx L1 -error estimate for monotone difference schemes
Mathematics of Computation
Numerical solution of the high frequency asymptotic expansion for the scalar wave equation
Journal of Computational Physics
Journal of Computational Physics
Multi-phase computations in geometrical optics
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
Mathematics of Computation
An Eulerian method for capturing caustics
Journal of Computational Physics
High-frequency wave propagation by the segment projection method
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
An Introduction to Eulerian Geometrical Optics (1992–2002)
Journal of Scientific Computing
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
A semiclassical transport model for two-dimensional thin quantum barriers
Journal of Computational Physics
High order numerical methods to a type of delta function integrals
Journal of Computational Physics
Journal of Computational Physics
High order numerical methods to two dimensional delta function integrals in level set methods
Journal of Computational Physics
Journal of Computational Physics
High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods
SIAM Journal on Scientific Computing
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In this paper, we construct two classes of Hamiltonian-preserving numerical schemes for a Liouville equation with discontinuous local wave speed. This equation arises in the phase space description of geometrical optics, and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics. We extend our previous work in [S. Jin, X. Wen, Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials, Commun. Math. Sci. 3 (2005) 285-315] for the semiclassical limit of the Schrodinger equation into this system. The designing principle of the Hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here, and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition. When a plane wave hits a flat interface, the Hamiltonian preservation is shown to be equivalent to Snell's law of refraction in the case when the ratio of wave length over the width of the interface goes to zero, when both length scales go to zero. Positivity, and stabilities in both l^1 and l^~ norms, are established for both schemes. The approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular (discontinuous and measure-valued) coefficients. Benchmark numerical examples are given, with analytic solution constructed, to study the numerical accuracy of these schemes.