The $l^1$-Stability of a Hamiltonian-Preserving Scheme for the Liouville Equation with Discontinuous Potentials

Abstract

We study the $l^1$-stability of a Hamiltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 (2005), 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the $l^1$-norm under a hyperbolic CFL condition which is in consistent with the $l^1$-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 (2008), 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become $l^1$-unstable.