The $L^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials and Perturbed Initial Data
Year: 2011
Journal of Computational Mathematics, Vol. 29 (2011), Iss. 1 : pp. 26–48
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 are 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1006-m3057
Journal of Computational Mathematics, Vol. 29 (2011), Iss. 1 : pp. 26–48
Published online: 2011-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Liouville equations Hamiltonian preserving schemes Piecewise constant potentials Error estimate Perturbed initial data Semiclassical limit.