SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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The Construction and Analysis of Variational Integrators

M. Leok and J. Hall

Abstract
Variational integrators are geometric integrators that are based on a discrete Hamilton's variational principle. We will review the role of Jacobi's solution of the Hamilton-Jacobi equation in the variational error analysis of variational integrators. Jacobi's solution can be characterized either in terms of a boundary-value problem or variationally, and these lead to shooting-based variational integrators and Galerkin variational integrators, respectively. Computable discrete Lagrangians can be obtained by choosing a numerical quadrature formula, and either a finite-dimensional function space or an underlying one-step method. We prove that the resulting variational integrator is order-optimal, and when spectral basis elements are used in the Galerkin formulation, one obtains geometrically convergent variational integrators. We will also discuss generalizations of variational integrators to Lie groups, homogeneous spaces and Lagrangian PDEs.

Bibliography
[1] J. Hall and M. Leok, Spectral Variational Integrators, arXiv: 1211.4534, 2012.
[2] J. Hall and M. Leok, Lie Group Galerkin Variational Integrators, in preparation, 2013.
[3] M. Leok and T. Shingel, General Techniques for Constructing Variational Integrators, Frontiers of Mathematics in China, 7(2), 273-303, 2012.
[4] J. Vankerschaver, C. Liao and M. Leok, Generating Functionals and Lagrangian PDEs, arXiv:1111.0280, 2012.

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