Mathematical Sciences: Immersed Interface Methods

许多重要的实际问题导致偏微分方程的解在某些界面上具有不连续或非光滑性。在数值求解这些问题时，使用均匀笛卡尔网格是非常方便的，尽管网格点之间的界面可能会切割。研究者和他的同事们开发了有限差分方法，旨在在所有网格点上给出高度精确的解决方案，同时保持统一网格方法的效率和易用性。对几个具体问题进行了深入研究。其中一个目标是开发改进版的佩斯金的“浸入边界法”，用于具有复杂移动边界区域的不可压缩流体动力学，例如跳动的心脏中的血流。在地震石油勘探中，为了模拟地球的结构，出现了带不连续系数的双曲波动方程。其他应用包括油藏模拟和地下水输送中出现的多孔介质方程，以及多相凝固问题。在模拟刚性边界附近的流体运动时，在求解边界处不连续强度的病态方程组时会出现一个附加问题。研究了求解这类问题的迭代方法。模拟现实世界中复杂结构的行为通常涉及求解无法精确求解的大型方程组。相反，必须在高性能计算机上用数值方法近似求解。研究人员研究的问题，其中有一个边界或界面，具有复杂的形状，并可能在时间运动。例子包括被液体包围的水体或气泡的表面，跳动的心脏的表面，或者油和注入地下以迫使石油从油田中流出的一些流体之间的边界。在这些例子中，所解的方程模拟了某种流体的流动。另一个问题是研究融化的冰和周围的水之间的边界运动，或者更一般地说，研究一种物质的不同相之间的边界运动。在这种情况下，方程模拟了热传导。在石油勘探中，有必要模拟地震波在地球中的运动，以及它们在地球深处不同岩石之间的界面上反射的方式。该项目的目标是开发相对简单的方法，可用于解决具有复杂边界或接口的各种此类问题。＊＊＊

9303404 LeVeque Many important practical problems lead to partial differential equations whose solutions have discontinuities or nonsmoothness across some interface. In solving these problems numerically, it is very convenient to use a uniform Cartesian grid in spite of the fact that the interface may cut between grid points. The investigator and his colleagues develop finite difference methods that aim to give highly accurate solutions at all grid points while maintaining the efficiency and ease of implementation of uniform grid methods. Several specific problems are studied in depth. One goal is to develop improved versions of Peskin's "immersed boundary method" for incompressible fluid dynamics in regions with complicated moving boundaries, such as blood flow in a beating heart. Hyperbolic wave equations with discontinuous coefficients arise in modeling the structure of the earth in seismic oil exploration. Other applications include porous media equations arising in oil reservoir simulation and groundwater transport, and multi-phase solidification problems. In modeling fluid motion near rigid boundaries, an additional problem arises in solving an ill-conditioned system of equations for the strength of discontinuities at the boundary. Iterative methods for solving such problems are studied. Simulating the behavior of complicated structures in the real world typically involves solving large systems of equations that cannot be solved exactly. Instead the solution must be approximated by numerical methods on high performance computers. The investigators study problems in which there is a boundary or interface that has a complicated shape and may be moving in time. Examples include the surface of a body of water or a bubble surrounded by fluid, the surface of a beating heart, or the boundary between oil and some fluid that is injected into the earth to force oil out of an oil field. In these examples the equations being solved model the flow o f some fluid. Another problem is to study the motion of the boundary between melting ice and the surrounding water, or between different phases of a substance more generally. In this case the equations model the conduction of heat. In oil exploration it is necessary to model the motion of seismic waves in the earth and the manner in which they reflect off interfaces between different kinds of rock deep within the earth. The goal of the project is to develop relatively simple methods that can be used to solve a wide variety of such problems with complicated boundaries or interfaces. ***