## 76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX)

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Lagrangian Coherent Structures (LCS) have become a widespread and powerful method to describe dynamic motion patterns in time-dependent flow fields. The standard way to extract LCS is to compute height ridges in the Finite Time Lyapunov Exponent (FTLE) field. In this work, we present an alternative method to approximate Lagrangian features for 2D unsteady flow fields that achieves subgrid accuracy without additional particle sampling. We obtain this by a geometric reconstruction of the flow map using additional material constraints for the available samples. In comparison to the standard method, this allows for a more accurate global approximation of LCS on sparse grids and for long integration intervals. The proposed algorithm works directly on a set of given particle trajectories and without additional flow map derivatives. We demonstrate its application for a set of computational fluid dynamic examples, as well as trajectories acquired by Lagrangian methods, and discuss its
benefits and limitations.

Recently, Holm and Ivanov, proposed and studied a class of multi-component
generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov,
Multi-component generalizations of the CH equation: geometrical
aspects, peakons and numerical examples,
{\it J. Phys A: Math. Theor} {\bf 43}, 492001 (20pp), 2010]. We
consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and
CH(2,2), and report a class of integrating factors and its
corresponding conservation laws for these two systems. In particular,
we obtain
the complete sent of first-order integrating factors for the systems
in Cauchy-Kovalevskaya form and evaluate the corresponding sets of
conservation laws for CH(2,1) and CH(2,2).