Volume 15, Issue 3
Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems

L. Brugnano

DOI:

J. Comp. Math., 15 (1997), pp. 233-252

Published online: 1997-06

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  • Abstract

In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called {\it symplectic integrators}. Several symplectic methods are known in the class of Runge-Kutta methods. However, no high order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are ``essentially'' maintained by the discrete map, at least for linear problems.

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@Article{JCM-15-233, author = {}, title = {Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems}, journal = {Journal of Computational Mathematics}, year = {1997}, volume = {15}, number = {3}, pages = {233--252}, abstract = { In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called {\it symplectic integrators}. Several symplectic methods are known in the class of Runge-Kutta methods. However, no high order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are ``essentially'' maintained by the discrete map, at least for linear problems. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9202.html} }
TY - JOUR T1 - Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems JO - Journal of Computational Mathematics VL - 3 SP - 233 EP - 252 PY - 1997 DA - 1997/06 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9202.html KW - AB - In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called {\it symplectic integrators}. Several symplectic methods are known in the class of Runge-Kutta methods. However, no high order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are ``essentially'' maintained by the discrete map, at least for linear problems.
L. Brugnano. (1970). Essentially Symplectic Boundary Value Methods for Linear Hamiltonian Systems. Journal of Computational Mathematics. 15 (3). 233-252. doi:
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