New results in numerical conservation laws

Prof. Pierre A. Gremaud

Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University

This talk is about the analysis of numerical methods for conservation laws. Such equations are ubiquitous in many applications, since conservation principles are the basis of countless models. In spite of this, the understanding of corresponding numerical methods is shockingly poor, even though efficient algorithms do exist. We shall focus on the origin of the difficulties encountered in the analysis of such methods. More specifically, three points are of importance: 1. poor regularity of the solutions (BV), 2. selection of the correct solutions, 3. effects of the mesh on the numerical solutions. A better understanding of those issues could lead not only to improved numerical methods, but also to the design of new algorithms for problems that do not fit into existing frameworks (e.g., mixed type problems). New results will be presented and discussed. This is joint work with B. Cockburn (University of Minnesota).