Numerical Methods for Moving Interfaces in Fluids

这项工作由两个项目组成，涉及具有运动界面的流体流动问题的数值方法。设计了一种改进的数值方法，用于粘性流体流动的联合运动，该运动由Navier-Stokes方程和对流体施加拉伸响应的移动边界共同控制。C.Peskin的浸没边界方法就是为这一原型问题发展起来的，并在生物学中有许多应用。在新的方法中，界面保持锐利，方法应该是二阶精度。速度可以分两部分得到：每次由界面力确定的定常斯托克斯速度，要么是从奇异积分中得到的，就像J·应变的相关工作中所做的那样，要么是从网格计算中得到的，比如浸没界面法。速度的其余规则部分将在矩形网格上使用时间倒退的特征进行计算。速度的分解使作用力集中在最需要作用力的界面上。对于初始发展，移动界面将由跟踪粒子来表示，但本方法可以与改进的界面运动方法相结合，例如应变的半拉格朗日等值线。将考虑隐式版本的方法，以避免由于界面运动而造成的时间步长限制。一名研究生将从事一项与曲面表示有关的项目。在工作的第二部分，我们将给出在界面上有间断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值进行了修正，就像R.Leveque和Z.Li的浸没界面方法以及A.Mayo的相关工作一样。这种方法允许用矩形网格的简单性来处理一般的界面边界。解的精度相对于截断误差的预期增益将取决于离散化的选择。这一选择将被调查，并将给出准确性的证明。许多科学问题涉及流体中的移动边界，例如一种液体在另一种液体中的滴落，或活组织中弹性膜的运动。对这类问题的数值研究有特殊的困难。在固定点上计算流体量是可取的，而不是依赖于移动边界的当前位置，但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样一种方法，用于已应用于几个生物学问题的原型模型。这样的改进可以扩大数值模拟在这些问题上的用处，因为它本质上更准确，因此对于大型计算，特别是三维计算，效率更高。第二个项目涉及对在存在边界的情况下使用差分运算符时所产生的误差的分析理解和估计，以及由此产生的不连续性。在用浸没界面法或本文提出的方法对具有运动边界的流体流动进行数值模拟时，需要这样的误差估计以确保近似的精度。

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.