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 - <p style="text-align: justify;">We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints.&nbsp; 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 $h&gt;0$, 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.</p>