AMMCS-2013 Semi-Plenary Talk

Boundary Conditions for Constrained Hyperbolic Systems: Mathematical and Numerical Analysis

Nicolae Tarfulea (Purdue University Calumet)

Abstract: Many applications in sciences and technology lead to first order symmetric hyperbolic (FOSH) systems of differential equations supplemented by constraint equations. The Cauchy problem for many such FOSH systems is constraint-preserving, i.e., the solution satis?es certain spatial differential constraints whenever the initial data does (e.g., Maxwell's equations or Einstein's field equations in various FOSH formulations). Frequently, artificial space cut offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artficial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Therefore, boundary conditions have to be posed in such a way that the numerical solution of the cut off system approximates as best as possible the solution of the original problem on infinite space, and this includes the preservation of constraints. It has become increasingly clear that in order for constraints to be preserved during evolution, the boundary conditions have to be chosen in an appropriate way. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal nonnegative boundary conditions. Based on a characterization of maximal nonnegative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by presenting a few relevant applications (e.g., for FOSH formulations of Einstein's equations and for systems of wave equations in FOSH formulation subject to divergence constraints).

Dr Nicolae Tarfulea is Associate Professor in the Department of Mathematics, Computer Science & Statistics, Purdue University Calumet. He received his PhD from the University of Minnesota and his M.S. in Mathematics from the Penn State University in 2004 and 2001, respectively. His main research Interests are in Partial Differential Equations; Numerical Analysis; General Relativity. More precisely: boundary conditions for hyperbolic formulations of Einstein's equations, nonlinear elliptic equations, reaction diffusion systems, compressed sensing, and finite element methods. He has published 23 papers on these subjects in some of the most prestigious journals of mathematics, and gave over 20 invited talks in the last five years, and been a part of six research grants.