An Efficient Lagrangian Interpolation Scheme for Computing Flow Maps and Line Integrals using Discrete Velocity Data

We propose a new class of Lagrangian approaches for constructing the flow maps of given dynamical systems. In the case when only discrete velocity data at mesh points is available, an interpolation step will be required. However, in our proposed approaches all particle trajectories share a common global interpolation at each time step and therefore interpolation operations will not increase the overall computational complexity. The old Lagrangian approaches propose to solve the corresponding ordinary differential equations (ODEs) backward in time to obtain the backward flow map. It is inconvenient and not natural, especially when incorporated with certain computational fluid dynamic solvers, because the velocity field needs to be loaded from the terminal time backward to the initial time. In contrast, our proposed approaches for computing the backward flow map propose to solve the corresponding ODEs forward in time which is more practical. We will also extend the proposed approaches to compute line integrals along any particle trajectory. This paper gives a detailed analysis on the computational complexity and error estimate of the proposed Lagrangian approach. Finally, a wide range of applications of our approaches will be given, including the so-called coherent ergodic partition and the high frequency wave propagations based on geometric optic.