An Eulerian method for computing the coherent ergodic partition of continuous dynamical systems

We develop an efficient Eulerian numerical approach to extract invariant sets in a continuous dynamical system in the extended phase space (the x- t space). We extend the idea of ergodic partition and propose a concept called coherent ergodic partition for visualizing ergodic components in a continuous flow. Numerically, we first apply the level set method [33] and extend the backward phase flow method [25] to determine the long time flow map. To compute all required long time averages of observables along particle trajectories, we propose an Eulerian approach by simply incorporating flow maps to iteratively interpolate those short time averages. Numerical experiments will demonstrate the effectiveness of the approach. As an application of the method, we apply the approach to the field of geometrical optics for high frequency wave propagation and propose to use the result from the coherent ergodic partition as a criteria for adaptivity in typical Lagrangian ray tracing methods.