When numerically solving the Liouville equation with a discontinuous potential, one
faces the problem of selecting a unique, physically relevant solution across the potential barrier, and
the problem of a severe time step constraint due to the CFL condition. In this paper, we introduce
two classes of Hamiltonian-preserving schemes for such problems. By using the constant Hamiltonian
across the potential barrier, we introduce a selection criterion for a unique, physically relevant
solution to the underlying linear hyperbolic equation with singular coefficients. These schemes have
a hyperbolic CFL condition, which is a signicant improvement over a conventional discretization.
These schemes are proved to be positive, and stable in both l∞ and l1 norms.
Numerical experiments are conducted to study the numerical accuracy. ¶
This work is motivated by the well-balanced kinetic schemes by Perthame and Simeoni for the
shallow water equations with a discontinuous bottom topography, and has applications to the level
set methods for the computations of multivalued physical observables in the semiclassical limit of
the linear Schr%#x000F6;dinger equation with a discontinuous potential, among other applications.