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 ^{1} 1 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, ChinaReceived 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 |