Eulerian Based Interpolation Schemes for Flow Map Construction and Line Integral Computation with Applications to Lagrangian Coherent Structures Extraction

We propose and analyze a new class of Eulerian methods for constructing both the forward and the backward flow maps of sufficiently smooth dynamical systems. These methods improve previous Eulerian approaches so that the computations of the forward flow map can be done on the fly as one imports or measures the velocity field forward in time. Similar to typical Lagrangian or semi-Lagrangian methods, the proposed methods require an interpolation at each step. Having said that, the Eulerian method interpolates d components of the flow maps in the d dimensional space but does not require any ( d + 1 ) -dimensional spatial-temporal interpolation as in the Lagrangian approaches. We will also extend these Eulerian methods to compute line integrals along any Lagrangian particle. The paper gives a computational complexity analysis and an error estimate of these Eulerian methods. The method can be applied to a wide range of applications for flow map constructions including the finite time Lyapunov exponent computations, the coherent ergodic partition, and high frequency wave propagations using geometric optic.