| This thesis focuses on the computation of high frequency waves, including elastic waves, and acoustic waves with weak random fluctuation, in heterogeneous media. The high frequency limits of these problems are described by Liouville equations or radiative transfer equations respectively, with discontinuous indices of refraction. The main idea is to extend the Hamiltonian-preserving schemes of Jin and Wen for semiclassical limit of Schrodinger equation with potential barriers, and geometrical optics through interfaces. In the case of elastic waves, the model conversion between the P-wave and the S-wave complicates the numerical method. In the case of radiative transfer equation, one needs to take into consideration both regular and diffuse reflections and transmissions. The essential idea of our method is to incorporate the interface transmission and reflection of waves into the numerical fluxes. By doing so we obtain numerical schemes that can (1) efficiently compute high frequency waves without numerically resolving the high frequencies; (2) capture the partial---including both regular and diffuse---transmission and reflection at the interfaces; (3) allow an optimal hyperbolic CFL condition.; Our methods are the first Eulerian methods for the problems under study. Numerous numerical experiments show the effectiveness and accuracy of the methods. In the case of the radiative transfer, we also show that the scheme, as the mean free path goes to zero, captures the correct diffusion limit with the corresponding interface conditions. |
