Shi Jin
Department of Mathematics, University of Wisconsin, Madison
When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of zero time step due to the CFL constraint, and the inconsistency to the constant Hamiltonian. In this paper, we propose a class of Hamiltonian-preserving schemes that are able to overcome these numerical deficiencies. The key idea is to build into the numerical flux the behavior of a classical particle at a potential barrier. We establish the stability theory of these new schemes, and analyze their numerical accuracy. Numerical experiments are carried out to verify the theoretical results. This method can also be applied to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schrodinger equation with a discontinuous potential, among other applications.