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Volume 41, Issue 1
Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

Lei Li & Dongling Wang

DOI: 10.4208/jcm.2106-m2020-0205

J. Comp. Math., 41 (2023), pp. 107-132.

Published online: 2022-11

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  • Abstract

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints.  The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.

  • Keywords

Hamiltonian systems, Holonomic constraints, symplecticity, Quadratic invariants, Partitioned Runge-Kutt methods.

  • AMS Subject Headings

65P10, 65L05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

leili2010@sjtu.edu.cn (Lei Li)

wdymath@nwu.edu.cn (Dongling Wang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-107, author = {Li , Lei and Wang , Dongling}, title = {Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {41}, number = {1}, pages = {107--132}, abstract = {

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints.  The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2106-m2020-0205}, url = {http://global-sci.org/intro/article_detail/jcm/21172.html} }
TY - JOUR T1 - Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints AU - Li , Lei AU - Wang , Dongling JO - Journal of Computational Mathematics VL - 1 SP - 107 EP - 132 PY - 2022 DA - 2022/11 SN - 41 DO - http://doi.org/10.4208/jcm.2106-m2020-0205 UR - https://global-sci.org/intro/article_detail/jcm/21172.html KW - Hamiltonian systems, Holonomic constraints, symplecticity, Quadratic invariants, Partitioned Runge-Kutt methods. AB -

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints.  The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.

Li , Lei and Wang , Dongling. (2022). Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints. Journal of Computational Mathematics. 41 (1). 107-132. doi:10.4208/jcm.2106-m2020-0205
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