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

Invited Talk

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Volume preserving numerical methods and generating forms

A. Zanna, H. Xue and O. Verdier

In this talk, we consider the problem of constructing volume-preserving maps (ultimately, numerical integrators) for divergence-free differential equations. We study the problem using the technique of differential forms and generating functions (generating forms), paying particular attention to the simplest nontrivial case that differs from the symplectic case, i.e. $n=3$. We propose a classification other than that of [1], who classified the generating equations, compatibility conditions and twist condition for volume forms. From our classification, we identify classes of generating forms that have a closed-form solution. Connections are made to the theory of discrete Lagrangians [2], and we identify a new type of generating equations that are not known in the context of symplectic maps.

[1] H. E. Lomelí and J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete and Continuous Dynamical Systems Series S, 2(2):361-377, 2009.
[2] J. E. Marsden and M. West. Discrete mechanics and variational integrators. Acta Numerica, 10:357-514, 2001.
[3] H. Xue and A. Zanna. Generating functions and volume-preserving mappings. To appear in DCDS-A, 2013.

Organized by         Universidad de Valladolid     IMUVA