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Volume 27, Issue 1
The $l^1$-Stability of a Hamiltonian-Preserving Scheme for the Liouville Equation with Discontinuous Potentials

Xin Wen & Shi Jin

J. Comp. Math., 27 (2009), pp. 45-67.

Published online: 2009-02

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  • Abstract

We study the $l^1$-stability of a Hamiltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 (2005), 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the $l^1$-norm under a hyperbolic CFL condition which is in consistent with the $l^1$-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 (2008), 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become $l^1$-unstable.

  • Keywords

Liouville equations, Hamiltonian preserving schemes, Discontinuous potentials, $l^1$-stability, Semiclassical limit.

  • AMS Subject Headings

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

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-45, author = {Xin Wen and Shi Jin}, title = {The $l^1$-Stability of a Hamiltonian-Preserving Scheme for the Liouville Equation with Discontinuous Potentials}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {1}, pages = {45--67}, abstract = {

We study the $l^1$-stability of a Hamiltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 (2005), 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the $l^1$-norm under a hyperbolic CFL condition which is in consistent with the $l^1$-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 (2008), 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become $l^1$-unstable.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10372.html} }
TY - JOUR T1 - The $l^1$-Stability of a Hamiltonian-Preserving Scheme for the Liouville Equation with Discontinuous Potentials AU - Xin Wen & Shi Jin JO - Journal of Computational Mathematics VL - 1 SP - 45 EP - 67 PY - 2009 DA - 2009/02 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10372.html KW - Liouville equations, Hamiltonian preserving schemes, Discontinuous potentials, $l^1$-stability, Semiclassical limit. AB -

We study the $l^1$-stability of a Hamiltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 (2005), 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the $l^1$-norm under a hyperbolic CFL condition which is in consistent with the $l^1$-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 (2008), 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become $l^1$-unstable.

Xin Wen and Shi Jin. (2009). The $l^1$-Stability of a Hamiltonian-Preserving Scheme for the Liouville Equation with Discontinuous Potentials. Journal of Computational Mathematics. 27 (1). 45-67. doi:
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