Numerical Methods for Moving Interfaces in Fluids

这项工作包括两个项目，涉及移动界面流体流动问题的数值方法。本文将设计一种改进的数值计算方法，用于计算由Navier-Stokes方程和运动边界控制的粘性流体的组合运动，该运动边界对流体的拉伸响应施加一个力。C. Peskin的浸入边界法是针对这一原型问题发展起来的，并在生物学中有许多应用。在新方法中，界面保持清晰，方法应具有二阶精度。速度将分为两部分：每次由界面力决定的稳定斯托克斯速度，将从奇异积分（如J. Strain的相关工作中所做的）或从网格计算（如浸入界面法）中得到。剩余的速度规则部分将在一个矩形网格上计算，使用时间向后的特征。速度的分解允许将努力集中在最需要的界面上。对于最初的发展，移动界面将由跟踪粒子来表示，但目前的方法可以与精细的界面运动方法相结合，如应变的半拉格朗日轮廓。该方法的隐式版本将被考虑，以避免由于接口运动的时间步长限制。研究生将在一个处理表面表示的相关项目上工作。在第二部分的工作中，将导出在界面处具有不连续的扩散方程的有限差分方法的最大误差估计，其中对界面附近的差异进行修正，如R. Leveque和Z. Li的浸入界面方法以及A. Mayo的相关工作。这样的方法允许用矩形网格的简单性来处理一般的界面边界。相对于截断误差的解中精度的预期增益将取决于离散化的选择。这个选择将被调查和准确性的证明将给出。许多科学问题涉及流体中的移动边界，例如一种流体在另一种流体中的一滴，或者活组织中弹性膜的运动。这类问题的数值研究具有特殊的困难。不依赖于移动边界的当前位置来计算固定点上的流体量是可取的，但必须考虑到跨越边界的数量的不连续。第一个项目的目的是改进这种方法的原型模型，该模型已应用于几个生物学问题。这种改进可以扩展数值模拟在这些问题中的有用性，因为它本质上更准确，因此对大型计算更有效，特别是在三维空间中。第二个项目涉及在存在边界的情况下使用差分算子时产生的误差的分析理解和估计，从而导致不连续性。这样的误差估计是需要的，以确保在使用浸入界面法或本文开发的方法对具有移动边界的流体流动进行数值模拟时所作近似的准确性。

The work consists of two projects concerning numerical methods for problems of fluid flow with moving interfaces. An improved numerical method will be designed for the combined motion of viscous fluid flow, governed by the Navier-Stokes equations, and a moving boundary which exerts a force on the fluid inresponse to its stretching. The immersed boundary method of C. Peskin was developed for this prototype problem and has had a number of applications in biology. In the new approach the interface is kept sharp and the method should be second-order accurate. The velocity will be found in two parts: at each time the steady Stokes velocity, determined by the interfacial force,will be found either from singular integrals, as done in related work with J. Strain, or from a grid calculation such as the immersed interface method. The remaining regular part of the velocity will be calculated on a rectangular grid using characteristics backward in time. The decomposition of the velocity allows effort to be concentrated at the interface, where it is most needed. For initial development the moving interface will be represented by tracking particles, but the present approach can be combined with refined methods for interface motion such as Strain's semi-Lagrangian contouring. Implicit versions of the method will be considered, to avoid time step limitations due to the interface motion. A graduate student will work on a related project dealing with the representation of surfaces. In the second part of the work,estimates of maximum errors will be derived for finite difference methods for diffusion equations with discontinuities at interfaces, in which corrections are added to the differences near the interface, as in the immersed interface method of R. Leveque and Z. Li and related work of A. Mayo. Such methods allow treatment of general interfacial boundaries with the simplicity of a rectangular grid. An expected gain of accuracy in the solution relative to the truncation error will depend on the choice of discretization. This choice will be investigated and proofs of accuracy will be given.A number of scientific problems involve moving boundaries in fluids, such as a drop of one fluid in another, or the motion of an elastic membrane in living tissue. Numerical study of such problems has special difficulties. It is desirable to calculate fluid quantities at fixed points not depending on the current location of the moving boundary, but discontinuities in quantities across the boundary must be taken into account. The aim in the first project is to improve such a method for a prototype model which has been applied to several biological problems. Such improvement could extend the usefulness of numerical simulation in these problems since it would be inherently more accurate and therefore more efficient for large computations, especiallyin three dimensions. The second project concerns analytical understanding and estimation of errors made when difference operators are used in the presence of boundaries, with resulting discontinuities. Such error estimates are needed to ensure the accuracy of approximations made in numerical simulation of fluid flow with moving boundaries using the immersed interface method or in the method developed here.