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Title: On numerical methods for Hamiltonian PDEs and a collocation method for the Vlasov-Maxwell equations

Abstract

Hamiltonian partial differential equations often have implicit conservation laws-constants of the motion-embedded within them. It is not, in general, possible to preserve these conservation laws simply by discretization in conservative form because there is frequently only one explicit conservation law. However, by using weighted residual methods and exploiting the Hamiltonian structure of the equations it is shown that at least some of the conservation laws are preserved in a method of lines (continuous in time). In particular, the Hamiltonian can always be exactly preserved as a constant of the motion. Other conservation laws, in particular linear and quadratic Casimirs and momenta, can sometimes be conserved too, depending on the details of the equations under consideration and the form of discretization employed. Collocation methods also offer automatic conservation of linear and quadratic Casimirs. Some standard discretization methods, when applied to Hamiltonian problems are shown to be derived from a numerical approximation to the exact Poisson bracket of the system. A method for the Vlasov-Maxwell equations based on Legendre-Gauss-Lobatto collocation is presented as an example of these ideas. 22 refs.

Authors:
 [1]
  1. Univ. of Michigan, Ann Arbor, MI (United States)
Publication Date:
OSTI Identifier:
478601
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 129; Journal Issue: 1; Other Information: PBD: Nov 1996
Country of Publication:
United States
Language:
English
Subject:
66 PHYSICS; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; HAMILTONIANS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; MAXWELL EQUATIONS

Citation Formats

Holloway, J P. On numerical methods for Hamiltonian PDEs and a collocation method for the Vlasov-Maxwell equations. United States: N. p., 1996. Web. doi:10.1006/jcph.1996.0238.
Holloway, J P. On numerical methods for Hamiltonian PDEs and a collocation method for the Vlasov-Maxwell equations. United States. https://doi.org/10.1006/jcph.1996.0238
Holloway, J P. 1996. "On numerical methods for Hamiltonian PDEs and a collocation method for the Vlasov-Maxwell equations". United States. https://doi.org/10.1006/jcph.1996.0238.
@article{osti_478601,
title = {On numerical methods for Hamiltonian PDEs and a collocation method for the Vlasov-Maxwell equations},
author = {Holloway, J P},
abstractNote = {Hamiltonian partial differential equations often have implicit conservation laws-constants of the motion-embedded within them. It is not, in general, possible to preserve these conservation laws simply by discretization in conservative form because there is frequently only one explicit conservation law. However, by using weighted residual methods and exploiting the Hamiltonian structure of the equations it is shown that at least some of the conservation laws are preserved in a method of lines (continuous in time). In particular, the Hamiltonian can always be exactly preserved as a constant of the motion. Other conservation laws, in particular linear and quadratic Casimirs and momenta, can sometimes be conserved too, depending on the details of the equations under consideration and the form of discretization employed. Collocation methods also offer automatic conservation of linear and quadratic Casimirs. Some standard discretization methods, when applied to Hamiltonian problems are shown to be derived from a numerical approximation to the exact Poisson bracket of the system. A method for the Vlasov-Maxwell equations based on Legendre-Gauss-Lobatto collocation is presented as an example of these ideas. 22 refs.},
doi = {10.1006/jcph.1996.0238},
url = {https://www.osti.gov/biblio/478601}, journal = {Journal of Computational Physics},
number = 1,
volume = 129,
place = {United States},
year = {Fri Nov 01 00:00:00 EST 1996},
month = {Fri Nov 01 00:00:00 EST 1996}
}