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. ***
