Higher order time discretization for free surface flows

在多相系统的数值近似中，或者更广泛地说，在处理具有毛细自由边界的流动时，基本问题是自由表面的表示、曲率的评估、不连续性的处理，最重要的是压力跳跃，以及流动计算和几何解耦的时间离散化策略。虽然对于前三个项目，存在着大量的文献和近二十年来发展起来的许多技术，但最后一个问题--如何有效地处理时间离散化--被广泛忽视。现有的大多数方法只是通过一种简单的分离方法将流场从几何中分离出来，即计算前一个时间步长的几何量。这些策略导致a)严重的毛细CFL条件，以及b)在时间上至多为一阶。现有的半隐式离散格式克服了问题a)，但仍然是一阶的，在某些情况下是相当耗散的。因此，本项目旨在开发、实施和分析无条件稳定且耗散最小的高阶时间离散化，从而允许相当大的时间步长。为了求解必然包含流场和几何之间的耦合的所产生的系统，将开发有效的求解技术。

The fundamental problems in the numerical approximation of multiphase systems, or more generally speaking, in the treatment of flows with capillary free boundaries are the representation of the free surface, evaluating the curvature, handling of discontinuities, most importantly the pressure jump, and time discretization strategies for the decoupling of flow computation and geometry. While for the first three items there exists a vast literature and many techniques developed over the last two decades, the last problem of how to efficiently treat the time discretization has been widely ignored. The majority of the existing approaches just decouple the flow field from the geometry by a simple segregated approach, i.e. evaluating the geometric quantities from the previous time step. These strategy leads to a) a severe capillary CFL condition and b) is of first order in time at most. Existing semi-implicit discretizations exist that overcome problem a), but are still first order only and rather dissipative in certain situations. Thus this project aims at developing, implementing, and analysing time discretizations of higher order that are unconditionally stable and minimally dissipative, thus allowing for rather large time steps. To solve the arising systems, which will necessarily comprise a coupling between flow field and geometry, efficient solution techniques will be developed.