On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems
The author introduces a new family of partitioned Runge-Kutta algorithms (PRK) for approximation to the solution of separable Hamiltonian systems with holonomic constraints. The new algorithms are based on modification of explicit PRK methods for unconstrained systems in the form given by \textit{D. Okunbor} and \textit{R. D. Skeel} [Math. Comput. 59, No. 200, 439-455 (1992; Zbl 0765.65079)]. They are semi-explicit and in applications require less computation than PRK programs in current use. An analysis is made of the order of error of the algorithms based on a more general study of the backward error for numerical integrators defined on a \(C^\infty\) manifold \(\mathcal M\) contained in \(\mathbb{R}^m\). It is proved that, except for orders one and two, the order of error in the constrained system may be less than the order of error of the corresponding unconstrained system. The author describes two extensions of the program which are designed so that the order of error of the constrained program is the same as the order of error of the unconstrained program. The first of these requires a special choice of coefficients in the approximating routine, and the second is based on composition of PRK algorithms which have error of first or second order. Numerical results are presented which result from applying the three programs to the equations for the motion of the double pendulum in a gravitational field. The effective order of error is found to be similar to error constants for unconstrained explicit PRK methods discussed by \textit{R. I. McLachlan} [SIAM J. Sci. Comput. 16, No. 1, 151-168 (1995; Zbl 0821.65048)].
semi-explicit symplectic partitioned Runge-Kutta methods
scientific article; zbMATH DE number 1027757