Mathematical Sciences: Immersed Interface Methods

研究者和他的同事们开发了浸入式界面方法，并将其应用于各种问题。浸入界面法是一类求解在某些界面上具有不连续或非光滑性的偏微分方程的方法。这个想法是使用一个统一的笛卡尔网格，尽管这个界面可能在网格点之间切割。这项工作的目标是开发二维和三维空间的有限差分方法，在保持统一网格方法的效率和易用性的同时，在所有网格点上给出高度精确的解。对于若干类型的问题，包括某些具有不连续系数的椭圆型、抛物型和双曲型方程，点态二阶精确方法已经得到了发展。同时还解决了运动界面下的斯托克斯流动和凝固问题。一些其他问题的扩展目前正在进行中。许多重要的实际问题导致了二维或三维空间区域的微分方程，这些区域在几何上是复杂的，并且包含解决方案性质变化的界面。这些方程很少能被精确地求解，并且需要大规模的计算才能在多维区域上得到很好的解。这项工作的目标是开发有效的计算方法来近似解决这类问题。具体研究了以下几个问题：(1)一个目标是基于广泛应用于生物物理和生理建模（如心脏、血管、内耳等）的Peskin方法，开发具有复杂移动弹性边界区域的不可压缩流体动力学改进方法。(2)非均质材料中的波动方程在许多应用中出现，包括超声成像和治疗、海洋声学和地球地震传播。“正演问题”是在给定材料性质和界面位置的情况下，及时正演求解方程。更难的“反问题”需要从表面测量中确定材料的特性和界面的位置。超声成像需要解决声学方程的反问题。例如，大量用于寻找石油的地球地震勘探需要解决弹性波动方程或简化旅行时间方程的逆问题。本文提出的方法既适用于正问题，也适用于逆问题。(3)在电阻抗层析成像中，解决了物体电导率的逆问题。这项技术用于医学成像，目前正在研究其目的是定位未爆地雷，这是当今许多国家面临的主要健康风险。这类逆问题在本工作中得到了研究。(4)油藏模拟和地下水输运过程中产生的多孔介质方程，需要在地质界面处重构渗透率和孔隙度的不连续面，以及代表油水界面的运动前沿等。准确的地下水流动模型对于研究有毒废物污染和污染场地的修复具有重要意义。(5)多相凝固问题涉及两相（如冰和水）之间的移动界面，并可能与流体动力学效应和传热相结合。实际应用包括研究洋流对冰架的影响、制造过程中的注射成型和化学蚀刻。

LeVeque 9626645 The investigator and his colleagues develop immersed interface methods and apply them to a variety of problems. Immersed interface methods are a class of methods for solving partial differential equations whose solutions have discontinuities or nonsmoothness across some interface(s). The idea is to use a uniform Cartesian grid in spite of the fact that the interface may cut between grid points. The goal of this work is to develop finite difference methods in two and three space dimensions that give highly accurate solutions at all grid points while maintaining the efficiency and ease of implementation of uniform grid methods. Pointwise second order accurate methods have already been developed for several classes of problems, including certain elliptic, parabolic, and hyperbolic equations with discontinuous coefficients across fixed interfaces. Stokes flow and solidification problems with moving interfaces have also been solved. Extensions to a number of other problems are currently underway. Many important practical problems lead to differential equations in regions of 2- or 3-dimensional space that are geometrically complicated, and that contain interfaces across which the nature of the solution changes. These equations can rarely be solved exactly, and large-scale computation is required to obtain well-resolved solutions over multi-dimensional regions. The goal of this work is to develop efficient computational methods to approximate solutions of such problems. Several specific problems are studied in depth: (1) One goal is to develop improved methods for incompressible fluid dynamics in regions with complicated moving elastic boundaries, based on a method of Peskin's that is widely used in biophysical and physiological modelling, e.g., of the heart, blood vesssels, the inner ear, etc. (2) Wave equations in heterogeneous materials arise in numerous applications, including ultrasound imaging and therapy, ocean acoustics, and seismic p ropagation in the earth. The "forward problem" consists of solving the equations forward in time given the material properties and locations of interfaces. The harder "inverse problem" requires determining the material properties and locations of interfaces from measurements made at the surface. Ultrasound imaging requires solving the inverse problem for acoustics equations. Seismic exploration of the earth, heavily used in the search for oil, for example, requires solving inverse problems for elastic wave equations or for simplified travel-time equations. The methods developed in this work are applied to both forward and inverse problems. (3) In electrical impedance tomography, an inverse problem is solved for the electrical conductivity of an object. This technique is used in medical imaging and is being studied for the purpose of locating unexploded land mines, a major health risk in many countries today. Inverse problems of this type are being studied in this work. (4) Porous media equations arising in oil reservoir simulation and groundwater transport requiremodeling discontinuities in permeability and porosity at geological interfaces, as well as moving front representing interfaces between oil and water, for example. Accurate modeling of groundwater flow is important in studying contamination by toxic wastes and remediation of contaminated sites. (5) Multi-phase solidification problems involve moving interfaces between the two phases (e.g. ice and water) and may be coupled with fluid dynamics effects as well as heat transfer. Practical applications include studying the effect of ocean currents on ice shelves, injection molding in manufacturing processes, and chemical etching.