Computational methods for coherent sets and coherent transport

湍流中的长寿命超结构可以被描述为潜在的随时间变化的速度场的所谓(拉格朗日)相干集。这些集合可以很好地计算为某个线性算子，即转移算子的本征函数的子级集。传统上，这种算子的离散化是基于空间中大量的拉格朗日轨迹和低阶ansatz函数的计算，导致即使是中等精度的小问题也需要相当昂贵的计算。最近，已经提出了使用高阶ansatz函数和相关的Fokker-Planck方程的直接时间积分的技术。这种方法允许在不计算拉格朗日轨迹的情况下计算拉格朗日相干集。然而，这些方法还不适用于对空间分辨率要求很高的三维湍流流动。在这个项目中，为了有效和快速地检测超结构，我们将开发新的数值技术来快速可靠地离散传递算子特征问题。这反过来将允许预测上层建筑的出现或分支，并可能允许对极端事件的短期预测。为了量化界面上的质量、热量和动量通量，我们将从拉格朗日的观点出发，进一步发展一种新的面向集合的三维流动输运方法，包括快速的数值实现。

Long living superstructures in turbulent fluids may be characterized as so-called (Lagrangian) coherent sets of the underlying time-dependent velocity field. These sets can be computed elegantly as sublevel sets of eigenfunctions of a certain linear operator, the transfer operator. Traditionally, the discretization of this operator has been based on the computation of a large number of Lagrangian trajectories and low order ansatz functions in space, resulting in rather expensive computations for even moderate accuracy and small problems. Recently, techniques have been proposed which use high-order ansatz functions and a direct time integration of the associated Fokker-Planck equation. This approach allows to compute Lagrangian coherent sets without computing Lagrangian trajectories. However, these methods are not yet applicable to 3D turbulent fluid flows which require a high spatial resolution.Within this project, in order to detect superstructures effectively and rapidly, we are going to develop new numerical techniques for a fast and reliable discretization of the transfer operator eigenproblem. This in turn will allow to predict the emergence or bifurcation of superstructures and potentially allow for short-term forecasts of extreme events. In order to quantify the fluxes of mass, heat and momentum across interfaces, we will further develop a new set-oriented methodology of transport in general 3D flows from the Lagrangian viewpoint, including a fast numerical implementation.