论文标题
能源和二次不变式保存具有自动限制的哈密顿系统的方法
Energy and quadratic invariants preserving methods for Hamiltonian systems with holonomic constraints
论文作者
论文摘要
我们为具有自动限制的哈密顿系统引入了一类新的参数化结构分区runge-kutta（$α$ -PRK）方法。当标量参数$α= 0 $时，这些方法将减少为通常的符号PRK方法，例如Shake-Rattle方法或基于Lobatto IIIA-IIIB对的PRK方案，该方法可以保留所有二次不变性和约束。当$α\ neq 0 $ $时，还显示了这些方法可以保留所有二次不变的且限制因素准确。同时，对于任何给定的一致初始值$（p_ {0}，q_0）$和小步骤$ h> 0 $，证明存在$α^*=α（h，p_0，q_0）$，使汉密尔顿能量在每个步骤中也可以完全保存。我们为Symbletic PRK方案提供了一种新的变异公式，并使用它来证明参数化的PRK方法可以保留受自动限制的二次不变性。参数$α$ -PRK方法显示出与通常的PRK方法相同的收敛速率，并且在各种数值实验中表现良好。
We introduce a new class of parametricization structure-preserving partitioned Runge-Kutta ($α$-PRK) methods for Hamiltonian systems with holonomic constraints. When the scalar parameter $α=0$, the methods are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs, which can preserve all the quadratic invariants and the constraints. When $α\neq 0$, the methods are also shown to preserve all the quadratic invariants and the constraints manifold exactly. At the same time, 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. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the parametrized PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. The parametric $α$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.
