Professor James Glimm and a group of U.S. researchers will conduct cooperative research with a group of Brazilian researchers under the leadership of Professor Dan Marchesin of the Pontificia Universidade Catolica of Rio de Janeiro. The investigators plan to pursue two basic scientific goals: to understand the nature of the fundamental solutions for the nonlinear partial differential equations representing conservation laws; and to devise an algorithm for the construction of these fundamental solutions and implement it as a computer program for systems with two dependent variables. This program will be used to perform highly accurate simulations of fluid flow in enhanced oil recovery. To optimize the recovery of hydrocarbons under several exploitation schemes, oil reservoir engineers use computer simulation to model the recovery process. The reservoir flow is described by a system of partial differential equations governing flow through a porous medium. Different phases or components of the fluid lead to discontinuities and numerical simulation of these flows often involves the solution of a system of conservation laws. At present there does not exist a body of theory or of numerical methods for solutions of these so-called Riemann problems in the large, and it is this need that the investigators plan to address. The collaborators have made substantial contributions to solving the Riemann problem for many systems of conservation laws and utilizing the solutions in applications. The theory and the computer programs to be developed in this joint work have broad potential application to gas dynamics, to combustion theory, to elastic and visco-elastic solids, and to high energy physics and astrophysics -- in fact, to any physical situation described by nonlinear conservation laws.
