Abstract: 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 introduced a selection criterion for a unique, physically relavant solution to the underlying linear hyperbolic equation with singular coefficients. These scheme have a hyperbolic CFL condition, which is a significant improvement over a conventional discretization. We also establish the positivity, and stability in both $l^1$ and $l^\infty$ norms, of these discretizations, and conducted numerical experiments 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 Schrödinger equation with a discontinuous potential, among other applications.
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