The l-error estimates for a Hamiltonian-preserving scheme to the Liouville equation with piecewise constant potentials

We study the l1-error of a Hamiltonian-preserving scheme, developed in [11], to the Liouville equation with a piecewise constant potential in one space dimension. This problem has important applications in computations of the semiclassical limit of the linear Schrödinger equation through barriers, and of the high frequency waves through interfaces. We use the l1-error estimates established in [28, 30] for the immersed interface upwind scheme to the linear advection equations with piecewise constant coefficients. We prove that the scheme with the Dirichlet incoming boundary conditions is l1-convergent for a class of bounded initial data, and derive the one-halfth order l1-error bounds with explicit coefficients. We show that the initial conditions can be satisfied by applying the decomposition technique proposed in [10] for solving the Liouville equation with measure-valued initial data, which arises in the semiclassical limit of the linear Schrödinger equation.