Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints (Journal of Computational Mathematics, 2022)
We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($α$-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter $α$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $α=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $α$-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 $α^*=α(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $α$-PRK methods. These $α$-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.