Numerical Methods for Moving Interfaces in Fluids

这项工作包括两个项目有关的数值方法与移动界面的流体流动问题。 将设计一种改进的数值方法，用于粘性流体流的组合运动，由Navier-Stokes方程控制，以及对流体施加力的移动边界，以响应其拉伸。 浸入边界法是C。Peskin是为这个原型问题而开发的，在生物学中有许多应用。 在新的方法中，界面保持尖锐，该方法应该是二阶精度。速度可以分为两部分：在每一时刻，由界面力决定的稳态斯托克斯速度可以通过奇异积分（如J. Strain的相关工作）或网格计算（如浸入界面法）来求出。 其余的常规部分的速度将计算一个矩形网格上使用的特点，在时间上。速度的分解使力集中在最需要的界面上。 对于初始开发的移动界面将表示跟踪粒子，但本方法可以结合界面运动的细化方法，如应变的半拉格朗日轮廓。 将考虑该方法的隐式版本，以避免由于界面运动而导致的时间步长限制。一个研究生将从事一个相关的项目，处理表面的表示。 在本文的第二部分中，我们将推导出在界面处具有间断的扩散方程的有限差分方法的最大误差估计，其中修正被加入到界面附近的差分中，就像R. Leveque和Z. Li和A.马约。这样的方法允许处理一般的界面边界与简单的矩形网格。相对于截断误差的解的预期精度增益将取决于离散化的选择。 许多科学问题涉及到流体中的运动边界，例如一种流体滴入另一种流体，或活组织中弹性膜的运动。 数值研究这类问题有特殊的困难。 理想的是计算不依赖于移动边界的当前位置的固定点处的流体量，但是必须考虑跨边界的量的不连续性。 第一个项目的目的是改进这种方法的原型模型已被应用到几个生物问题。 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，因此对于大型计算，特别是三维计算更有效。 第二个项目涉及的分析理解和估计的错误时，差分算子使用的存在的边界，所产生的不连续性。需要这样的误差估计，以确保在流体流动的数值模拟与移动边界的浸没界面法或在这里开发的方法中的近似的准确性。

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.