Symplectic Partitioned Runge–Kutta Methods for Constrained Hamiltonian Systems

Abstract

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.

Geometric Generalisations of shake and rattle

“…One would like to integrate an index-K DAE on M or an index-K + 1 DAE on a symplectic embedding of M so as to preserve the constraints and ı M K ω. Finally, we mention another class of integrators for the holonomic case, known as spark, for Symplectic Partitioned Additive Runge-Kutta [9]. These generalise RATTLE to higher order.…”

“…SHAKE and RATTLE are two commonly used numerical integration methods for Hamiltonian problems subject to holonomic constraints [16,1,11,9,15]. The difference between the two methods is that RATTLE preserves "hidden" constraints, whereas SHAKE does not.…”

On the multisymplecticity of partitioned Runge–Kutta and splitting methods

“…(20) as well, although the differential forms are evaluated on solutions of Eq. (48) instead of (19). One can derive a DMSCL as well, although there seems to be no unique or natural way to do this.…”

“…They were originally proposed for use in constrained systems, to which GRK did not appear to apply [19]. An exception is the lowest order Lobatto IIIA-IIIB or "generalized leapfrog" method, which has somewhat simpler implicit equations than the higher order methods.…”

Equivariant constrained symplectic integration

“…This preserves the standard symplectic form restricted to the constraint surface. There are other algorithms that also do this: for simple mechanical systems (when the Hamiltonian is the sum of a quadratic kinetic energy and a potential), the "RATTLE" method of molecular dynamics [2] was shown to be symplectic in this sense by Leimkuhler and Skeel [11]; Jay [10] gives a class of suitable partitioned Runge-Kutta methods, one of which reduces to RATTLE in the case of a simple mechanical system. The problem for more general constraints is still open.…”