Numerische Mathematik

, Volume 130, Issue 4, pp 681–740 | Cite as

Spectral variational integrators

  • James Hall
  • Melvin LeokEmail author


In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are arbitrarily high-order. In particular, if the quadrature formula used is sufficiently accurate, then the resulting Galerkin variational integrator has a rate of convergence at the discrete time-steps that is bounded below by the approximation order of the finite-dimensional function space. In addition, we show that the continuous approximating curve that arises from the Galerkin construction converges on the interior of the time-step at half the convergence rate of the solution at the discrete time-steps. We further prove that certain geometric invariants also converge with high-order, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence.

Mathematics Subject Classification

37M15 65M70 65P10 70H25 



We are very grateful to the referee of this paper, whose comments and observations significantly improved our exposition. This work was supported in part by NSF Grants CMMI-1029445, DMS-1065972, CMMI-1334759, DMS-1411792, and NSF CAREER Award DMS-1010687.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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