The L<sup>1</sup>-Error Estimates for a Hamiltonian-preserving Scheme for the Liouville Equation with Piecewise Constant Potentials and Perturbed Initial Data

Abstract

We study the L 1 -error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation errors. We extend the l 1 -stability analysis in [46] and apply the L 1 -error estimates with exact initial data established in [45] for the same scheme. We prove that the scheme with the Dirichlet incoming boundary conditions and for a class of bounded initial data is L 1 -convergent when the initial data is given with a wide class of perturbation errors, and derive the L 1 -error bounds with explicit coefficients. The convergence rate of the scheme is shown to be less than the order of the initial perturbation error, matching with the fact that the perturbation solution can be l 1 -unstable.