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Volume 29, Issue 1
The $L^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials and Perturbed Initial Data

Xin Wen

DOI: 10.4208/jcm.1006-m3057

J. Comp. Math., 29 (2011), pp. 26-48.

Published online: 2011-02

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  • 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.

  • Keywords

Liouville equations, Hamiltonian preserving schemes, Piecewise constant potentials, Error estimate, Perturbed initial data, Semiclassical limit.

  • AMS Subject Headings

65M06, 65M12, 65M25, 35L45, 70H99.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-29-26, author = {Xin Wen}, title = {The $L^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials and Perturbed Initial Data}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {1}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1006-m3057}, url = {http://global-sci.org/intro/article_detail/jcm/8462.html} }
TY - JOUR T1 - The $L^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials and Perturbed Initial Data AU - Xin Wen JO - Journal of Computational Mathematics VL - 1 SP - 26 EP - 48 PY - 2011 DA - 2011/02 SN - 29 DO - http://doi.org/10.4208/jcm.1006-m3057 UR - https://global-sci.org/intro/article_detail/jcm/8462.html KW - Liouville equations, Hamiltonian preserving schemes, Piecewise constant potentials, Error estimate, Perturbed initial data, Semiclassical limit. AB -

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.

Xin Wen. (2011). The $L^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials and Perturbed Initial Data. Journal of Computational Mathematics. 29 (1). 26-48. doi:10.4208/jcm.1006-m3057
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