Construction and analysis of higher order Galerkin variational integrators
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Construction and analysis of higher order Galerkin variational integrators Sina Ober-Bl¨obaum · Nils Saake
Received: 4 April 2014 / Accepted: 22 October 2014 © Springer Science+Business Media New York 2014
Abstract In this work we derive and analyze variational integrators of higher order for the structure-preserving simulation of mechanical systems. The construction is based on a space of polynomials together with Gauss and Lobatto quadrature rules to approximate the relevant integrals in the variational principle. The use of higher order schemes increases the accuracy of the discrete solution and thereby decrease the computational cost while the preservation properties of the scheme are still guaranteed. The order of convergence of the resulting variational integrators is investigated numerically and it is discussed which combination of space of polynomials and quadrature rules provide optimal convergence rates. For particular integrators the order can be increased compared to the Galerkin variational integrators previously introduced in Marsden and West (Acta Numerica 10:357–514 2001). Furthermore, linear stability properties, time reversibility, structure-preserving properties as well as efficiency for the constructed variational integrators are investigated and demonstrated by numerical examples. Keywords Discrete variational mechanics · Numerical convergence analysis · Symplectic methods · Variational integrators Mathematics Subject Classifications (2010) 37J99 · 65Lxx · 70H25
Communicated by: Axel Voigt S. Ober-Bl¨obaumm () · N. Saake Computational Dynamics and Optimal Control, Department of Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany e-mail: [email protected]
S. Ober-Bl¨obaum, N. Saake
1 Introduction During the last years the development of geometric numerical integrators has been of high interest in numerical integration theory. Geometric integrators are structurepeserving integrators with the goal to capture the dynamical system’s behavior in a most realistic way [7, 20, 27]. Using structure-preserving methods for the simulation of mechanical systems, specific properties of the underlying system are handed down to the numerical solution, for example, the energy of a conservative system shows no numerical drift or the first integrals induced by symmetries are preserved exactly. One particular class of structure-preserving integrators is the class of variational integrators, introduced in [20] and [33] and which has been further developed and extended to different systems and applications during the last years. Variational integrators [20] are based on a discrete variational formulation of the underlying system, e.g. based on a discrete version of Hamilton’s principle for conservative mechanical systems. The resulting integrators are symplectic and momentum-preserving and have an excellent long-time energy behavior. By choosing different variational formulations (e.g. Hamilton, Lagranged’Alembert, Hamilton-Pontryagin, etc.), variational integrators have been de
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