We propose an Eulerian approach to compute the expected finite time Lyapunov exponent (FTLE) of uncertain flow fields. The definition extends the usual FTLE for deterministic dynamical systems. Instead of performing Monte Carlo simulations as in typical Lagrangian computations, our approach associates each initial flow particle with a probability density function (PDF) which satisfies an advection-diffusion equation known as the Fokker-Planck (FP) equation. Numerically, we incorporate Strang's splitting scheme so that we can obtain a second-order accurate solution to the equation. To further improve the computational efficiency, we develop an adaptive approach to concentrate the computation of the FTLE near the ridge, where the so-called Lagrangian coherent structure (LCS) might exist. We will apply our proposed algorithm to several test examples including a real-life dataset to demonstrate the performance of the method.

我们提出一种欧拉方法来计算期望值不确定时间流场的有限时间Lyapunov指数（FTLE）。该定义扩展了确定性动力系统的常规FTLE。我们的方法不是像典型的拉格朗日计算中那样执行蒙特卡罗模拟，而是将每个初始流动粒子与概率密度函数（PDF）关联，该概率密度函数满足被称为福克-普朗克（FP）方程的对流扩散方程。在数值上，我们结合了Strang的分裂方案，以便可以对该方程获得二阶精确解。为了进一步提高计算效率，我们开发了一种自适应方法，将FTLE的计算集中在可能存在所谓的拉格朗日相干结构（LCS）的山脊附近。