A classification of volume preserving generating forms in $\mathbb{R}^3$

Olivier  Verdier; Huiyan  Xue; Antonella Zanna

doi:10.3934/dcds.2016.36.2285

Article Contents

2016, Volume 36, Issue 4: 2285-2303. Doi: 10.3934/dcds.2016.36.2285

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A classification of volume preserving generating forms in $\mathbb{R}^3$

1.
Department of Computing, Mathematics and Physics, Bergen University College, 5063 Bergen
2.
Department of Mathematics, University of Bergen, 5020 Bergen
3.
Department of Mathematics, University of Bergen, P.O. Box 7800, N-5020 Bergen

Received: December 2014

Revised: July 2015

Published online: September 2015

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Abstract

In earlier work, Lomeli and Meiss [9] used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the $\mathbb{R}^3$ case, the first to differ from the symplectic case, they derived thirty-six one-forms that generate exact volume preserving maps. In [20], Xue and Zanna studied these differential forms in connection with the numerical solution of divergence-free differential equations: can such forms be used to devise new volume preserving integrators or to further understand existing ones? As a partial answer to this question, Xue and Zanna showed how six of the generating volume form were naturally associated to consistent, first order, volume preserving numerical integrators. In this paper, we investigate and classify the remaining cases. The main result is the reduction of the thirty-six cases to five essentially different cases, up to variable relabeling and adjunction. We classify these five cases, identifying two novel classes and associating the other three to volume preserving vector fields under a Hamiltonian or Lagrangian representation. We demonstrate how these generating form lead to consistent volume preserving schemes for volume preserving vector fields in $\mathbb{R}^3$.

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Mathematics Subject Classification: Primary: 53D22, 70H15, 65L05.

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