We develop a class of Hamiltonian-preserving numerical schemes for high frequency elastic waves in heterogeneous media. The approach is based on the high frequency approximation governed by the Liouville equations with singular coefficients due to material interfaces. As previously done by Jin and Wen [10,12], we build into the numerical flux the wave scattering information at the interface, and use the Hamiltonian preserving principle to couple the wave numbers at both sides of the interface. This gives a class of numerical schemes that allows a hyperbolic CFL condition, is positive and l ∞ stable, and captures correctly wave scattering at the interface with a sharp numerical resolution. We also extend the method to curved interfaces. Numerical experiments are carried out to study the numerical convergence and accuracy.

In this paper, we numerically study the semiclassical limit of the Schrödinger-Poisson equations as a selection principle for the weak solution of the VlasovPoisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker-Planck equation. We also numerically justify the multivalued solution given by a moment system of the Vlasov-Poisson equations as the semiclassical limit of the Schrödinger-Poisson equations.

In this paper, we develop a numerical scheme for the interface problem in the planar symmetric radiative transfer equation with isotropic scattering. Such problems arise in the modeling of the propagation of energy density for waves in heterogeneous media with weak random fluctuation in the high frequency regime. The idea, following the earlier work of Jin and Wen for regular transmission and reflection, is to build the interface condition, which characterizes the reflection and transmission, into the numerical flux. The new contribution of this article is to deal with the diffuse transmission at the interface. The positivity of the scheme is proven. Moreover, we show, via numerical examples, that this new scheme is able to capture the correct (regular or diffuse) transmission and reflection through the interface, and, as the mean free path goes to zero, captures the diffusion limit with the correct interface conditions.