Mathematical Sciences: Immersed Interface Methods

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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.