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Volume 35, Issue 6
$ℓ^1$-Error Estimates on the Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials: A Simple Proof

Xinchun Li

DOI: 10.4208/jcm.1701-m2016-0717

J. Comp. Math., 35 (2017), pp. 814-827.

Published online: 2017-12

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  • Abstract

This work is concerned with $ℓ^1$-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with piecewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The $ℓ^1$-error estimates are then evaluated by comparing the derived equations and schemes.

  • Keywords

Liouville equations, Hamiltonian-preserving schemes, Piecewise constant potentials, $ℓ^1$-error estimate, Half-order error bound, Semiclassical limit.

  • AMS Subject Headings

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

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

neo.lee@sjtu.edu.cn (Xinchun Li)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-814, author = {Li , Xinchun}, title = {$ℓ^1$-Error Estimates on the Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials: A Simple Proof}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {6}, pages = {814--827}, abstract = {

This work is concerned with $ℓ^1$-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with piecewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The $ℓ^1$-error estimates are then evaluated by comparing the derived equations and schemes.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1701-m2016-0717}, url = {http://global-sci.org/intro/article_detail/jcm/10496.html} }
TY - JOUR T1 - $ℓ^1$-Error Estimates on the Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials: A Simple Proof AU - Li , Xinchun JO - Journal of Computational Mathematics VL - 6 SP - 814 EP - 827 PY - 2017 DA - 2017/12 SN - 35 DO - http://doi.org/10.4208/jcm.1701-m2016-0717 UR - https://global-sci.org/intro/article_detail/jcm/10496.html KW - Liouville equations, Hamiltonian-preserving schemes, Piecewise constant potentials, $ℓ^1$-error estimate, Half-order error bound, Semiclassical limit. AB -

This work is concerned with $ℓ^1$-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with piecewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The $ℓ^1$-error estimates are then evaluated by comparing the derived equations and schemes.

Li , Xinchun. (2017). $ℓ^1$-Error Estimates on the Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials: A Simple Proof. Journal of Computational Mathematics. 35 (6). 814-827. doi:10.4208/jcm.1701-m2016-0717
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