Eulerian Methods for Visualizing Continuous Dynamical Systems using Lyapunov Exponents

Abstract

We propose a new Eulerian numerical approach for constructing the forward flow maps in continuous dynamical systems. The new algorithm improves the original formulation developed in [23,24] so that the associated partial differential equations (PDEs) are solved forward in time and, therefore, the forward flow map can now be determined on the fly. Due to the simplicity in the implementations, we are now able to efficiently compute the unstable coherent structures in the flow based on quantities like the finite time Lyapunov exponent (FTLE), the finite size Lyapunov exponent (FSLE) and also a related infinitesimal size Lyapunov exponent (ISLE). When applied to the ISLE computations, the Eulerian method is in particularly computational efficient. For each separation factor r in the definition of the ISLE, typical Lagrangian methods require to shoot and monitor an individual set of ray trajectories. If the scale factor in the definition is changed, these methods have to restart the whole computations all over again. The proposed Eulerian method, however, requires to extract only an isosurface of a volumetric data for an individual value of r which can be easily done using any well-developed efficient interpolation method or simply an isosurface extraction algorithm. Moreover, we provide a theoretical link between the FTLE and the ISLE fields which explains the similarity in these solutions observed in various applications.

Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media

“…In this work, we apply the fifth-order weighted essentially non-oscillatory (WENO)-Godunov scheme (Jiang and Peng, 2000) for approximating the spatial derivatives and use a third-order totalvariation diminishing (TVD)-Runge-Kutta method (Osher and Shu, 1991) for time stepping. One might also consider the flow-map based methods developed in You et al (2017); Leung (2018, 2020).…”

Computing the finite time Lyapunov exponent for flows with uncertainties

“…We are not able to quantitatively compare our solution with any exact solution but can only qualitatively examine the effect of D 0 on our numerical solution. In Figure 17, we reproduce Figure 11 from [47] or Figure 7(a) from [46] showing the FTLE e 50 0 for the noiseless velocity, i.e. D 0 = 0.…”

“…Following the definition of Haller [13,15,16], one can see that the LCS is closely related to the ridges of the FTLE fields. In a series of recent studies [25,26,44,47,46,45,27,32], we have developed various Eulerian approaches to numerically compute the FTLE on a fixed Cartesian mesh. The idea is to combine the approach with the level set method [34,33] which allows the flow map to satisfy a Liouville equation.…”

“…Figure 17: (Section 5.2) The FTLE e 50 0 for the velocity without uncertainty, i.e. D 0 = 0, as shown in Figure11from[47] or Figure7(a) from[46].…”