Abstract: When
numerically solving the Liouville equation with a
discontinuous potential, one faces the problem of
severe time step restriction, and the inconsistency
to the constant Hamiltonian which is related to the
problem of how the weak solution should be defined
for such linear hyperbolic equations with singular
coefficients. In this talk, we present a class of
Hamiltonianpreserving 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 Schrödinger
equation with a discontinuous potential. For wave
equations with discontinuous local speeds, this
leads to numerical schemes consistent with Snell's
Law of Refraction.
[LECTURE SLIDES]
