This article deals with the numerical treatment of Hamiltonian systems with holonomic constraints.A class of partitioned Runge-Kutta methods, consisting of the couples of s-stage Lobatto IIIA and Lobatto IIIB methods, has been discovered to solve these problems efficiently. These methods are symplectic, preserve all underlying constraints, and are superconvergent with order 2s-2. For separable Hamiltonians of the form H (q, p) 1 / 2 prM-p + U (q) the Rattle algorithm based on the Verlet method was up to now the only known symplectic method preserving the constraints. In fact this method turns out to be equivalent to the 2-stage Lobatto IIIA-IIIB method of order 2. Numerical examples have been performed which illustrate the theoretical results.

One of the most difficult problems in the numerical solution of ordinary differential equations (ODEs) and in differential-algebraic equations (DAEs) is the development of methods for dealing with highly oscillatory systems. These types of systems arise, for example, in vehicle simulation when modelling the suspension system or tyres, in models for contact and impact, in flexible body simulation from vibrations in the structural model, in molecular dynamics, in orbital mechanics, and in circuit simulation. Standard numerical methods can require a huge number of time-steps to track the oscillations, and even with small stepsizes they can alter the dynamics, unless the method is chosen very carefully.

A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler-Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge-Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-CC * SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.

We consider a general class of systems of overdetermined differential-algebraic equations (ODAEs). We are particularly interested in extending the application of the symplectic Gauss methods to Hamiltonian and Lagrangian systems with holonomic constraints. For the numerical approximation to the solution to these ODAEs, we present specialized partitioned additive Runge-Kutta (SPARK) methods, and in particular the new class of (s, s)-Gauss-Lobatto SPARK methods. These methods not only preserve the constraints, symmetry, symplecticness of the flow, and variational nature of the trajectories of holonomically constrained Hamiltonian and Lagrangian systems, but they also have an optimal order of convergence 2s.

No abstract