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.
"Hamiltonian-Preserving Schemes for the Liouville Equation with Discontinuous Potentials." Commun. Math. Sci. 3 (3) 285 - 315, September 2005.