J. Comp. Math., 28 (2010), pp. 939-961. The $L^1$-Error Estimates for a Hamiltonian-preserving Scheme for the Liouville Equation with Piecewise Constant Potentials and Perturbed Initial Data Xin Wen 11 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Received 2009-2-23 Accepted 2010-4-30 Available online 2010-9-20 doi:10.4208/jcm.1004-m3057 Abstract We study the $L^1$-error of a Hamiltonian-preserving scheme, developed in \cite{JW}, 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 \cite{WJ} and apply the $L^1$-error estimates with exact initial data established in \cite{WJ3} 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 {\it 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. Key words: Liouville equations, Hamiltonian preserving schemes, Piecewise constant potentials, Error estimate, Perturbed initial data, Semiclassical limit. AMS subject classifications: 65M06, 65M12, 65M25, 35L45, 70H99. Email: wenxin@amss.ac.cn