Development and Analysis of Numerical Methods for Fluid Interfaces

本文研究粘性流体流动中材料界面运动的数值方法。这种模型特别用于小尺度的生物过程，其中界面对流体施加力。要做的工作将集中在表面积分的有效计算代表粘度占主导地位的流动，或斯托克斯流，和误差分析有限差分方法更一般的纳维尔-斯托克斯流，如浸入界面法，其中方程离散化的规则网格，修改界面附近的帐户的力量。所考虑的方法应至少具有二阶精度。在第一个项目中，将开发一种相对简单的方法来计算光滑表面上的奇异或近似奇异积分，例如三维Stokes流的速度积分，在表面上或表面附近进行计算。这项工作将改进和推广早期的工作，其中一个标准的积分的正则化结合校正附近的奇异性分析发现。即使对于曲面附近的网格点，该方法也应该是准确的，从而在计算曲面的运动时具有更大的灵活性。它可以被用作计算Navier-Stokes流的一部分。在第二个项目中，误差估计的最大范数将推导出有限差分方法，如浸入界面法。最近的结果的提议者显示增益的规律性的泊松或扩散方程的有限差分版本将被用来澄清之间的关系的数值解的精度和附近的接口所需的校正，也选择时间离散化。工作将包括收敛性证明简化界面问题的Navier-Stokes流和最大范数稳定性的近似投影的发散自由向量场。许多科学问题涉及到流体中的移动边界，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，因为它是可取的离散流体变量在一个固定的网格，而移动的边界必须单独表示，连同其对流体运动的影响。用一种简单实用的方法对运动曲面进行离散化是一件困难的事情。对于Stokes流，主要由粘度，流体变量的积分公式被广泛使用。所提出的积分方法有望比标准方法更简单，更有效，并且需要更少的表面工作。因此，随着模型变得更加逼真，它可以有助于三维模拟的实用性。第二个项目强调最大误差的估计，因为这些可能是最大的界面附近。他们是欠发达的估计比整体规范。这种估计将用于浸入界面法、上述纳维尔-斯托克斯流的分解和近似投影法等方法。现有的数值方法的误差分析，应提高其有效性和局限性的理解。它还可以显示方法中的选择如何影响其准确性并提出改进建议。

This work concerns numerical methods for material interfaces moving in viscous fluid flow. Such models are used especially for biological processes on small scales, in which the interface exerts a force on the fluid. The work to be done will focus on the efficient computation of surface integrals representing viscosity-dominated flow, or Stokes flow, and error analysis of finite difference methods for more general Navier-Stokes flow, such as the immersed interface method, in which the equations are discretized on a regular grid, with modifications near the interface to account for the forces. The methods considered should be at least second-order accurate. In the first project a relatively simple method will be developed for computing singular or nearly singular integrals on smooth surfaces, such as the velocity integrals for three-dimensional Stokes flow, evaluated on or near the surface. This work will improve and generalize earlier work, in which a standard quadrature of a regularized integral is combined with corrections found by analysis near the singularity. This method should be accurate even for grid points near the surface, allowing more flexibility in computing the motion of the surface. It could be used as part of a computation for Navier-Stokes flow. In the second project, error estimates in maximum norm will be derived for finite difference methods such as the immersed interface method. Recent results of the proposer showing a gain in regularity for finite difference versions of Poisson or diffusion equations will be used to clarify the relationship between the accuracy of numerical solutions and the corrections needed near the interface and also the choice of time discretization. The work will include convergence proofs for simplified interface problems with Navier-Stokes flow and maximum norm stability of the approximate projection on divergence-free vector fields. 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, since it is desirable to discretize the fluid variables on a fixed grid, while the moving boundary must be represented separately, together with its influence on the fluid motion. It is difficult to discretize the moving surface in a way that is simple and practical. For Stokes flow, dominated by viscosity, integral formulations of the fluid variables are widely used. The proposed method of integration promises to be simpler and more efficient than standard methods and require less effort with the surface. Thus it could contribute to the practicality of three-dimensional simulations as the models become more realistic. The second project emphasizes estimates of maximum errors, since these are likely to be largest near interfaces. They are less well developed than estimates in integral norms. Such estimates will be used for methods such as the immersed interface method, the decomposition of Navier-Stokes flow mentioned above, and approximate projection methods. Error analysis of existing numerical methods should improve understanding of their validity and limitations. It can also show how choices in the methods affect their accuracy and suggest improvements.