DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods

We present an algorithm for the computation of finite-time Lyapunov exponent (FTLE) fields using discontinuous-Galerkin (dG) methods in two dimensions. The algorithm is designed to compute FTLE fields simultaneously with the time integration of dG-based flow solvers of conservation laws. Fluid tracers are initialized at Gauss-Lobatto quadrature nodes within an element. The deformation gradient tensor, defined by the deformation of the Lagrangian flow map in finite time, is determined per element with high-order dG operators. Multiple flow maps are constructed from a particle trace that is released at a single initial time by mapping and interpolating the flow map formed by the locations of the fluid tracers after finite time integration to a unit square master element and to the quadrature nodes within the element, respectively. The interpolated flow maps are used to compute forward-time and backward-time FTLE fields at several times using dG operators. For a large finite integration time, the interpolation is increasingly poorly conditioned because of the excessive subdomain deformation. The conditioning can be used in addition to the FTLE to quantify the deformation of the flow field and identify subdomains with material lines that define Lagrangian coherent structures. The algorithm is tested on three benchmarks: an analytical spatially periodic gyre flow, a vortex advected by a uniform inviscid flow, and the viscous flow around a square cylinder. In these cases, the algorithm is shown to have spectral convergence.