American Institute of Mathematical Sciences

April  2016, 36(4): 2285-2303. doi: 10.3934/dcds.2016.36.2285

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

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

Received  December 2014 Revised  July 2015 Published  September 2015

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 $\mathbb{R}^3$.
Citation: Olivier Verdier, Huiyan Xue, Antonella Zanna. A classification of volume preserving generating forms in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2285-2303. doi: 10.3934/dcds.2016.36.2285
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