Computational methods for coherent sets and coherent transport

湍流流体中的长寿超结构可以被表征为潜在的时间相关的速度场的所谓的（拉格朗日）相干集。 这些集合可以被优雅地计算为某个线性算子（转移算子）的本征函数的子级集合。 传统上，该算子的离散化是基于空间中大量拉格朗日轨迹和低阶反函数的计算，即使是中等精度和小问题也会导致相当昂贵的计算。 最近，人们提出了使用高阶拟函数和相关福克-普朗克方程的直接时间积分的技术。 这种方法允许计算拉格朗日相干集，而无需计算拉格朗日轨迹。 然而，这些方法还不适用于三维湍流流体流动，需要高的空间分辨率，在这个项目中，为了有效地检测上层建筑，迅速，我们将开发新的数值技术的快速和可靠的离散化的传输算子特征问题。 这反过来将有助于预测上层建筑的出现或分叉，并可能有助于对极端事件进行短期预测。 为了量化跨接口的质量，热量和动量的通量，我们将进一步开发一种新的面向集合的方法，从拉格朗日的观点，包括快速的数值实现一般的3D流的运输。

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.