Numerical Analysis Seminar

Wednesday, March 21, 2007

Speaker:	Tim Barth, NASA Ames Research Center
Title:	Energy Stable FEM Discretization of Nonlinear Conservation Laws: A Contrasting Look at Hydrodynamics and Magnetohydrodynamics
Time:	3:00-4:00 pm
Place:	Blocker 628

Abstract

A self-contained energy analysis is briefly outlined for the discontinuous Galerkin (DG) discretization [1] applied to the compressible magnetohydrodynamic (MHD) equations with solenoidally constrained magnetic induction field, div B = 0. This analysis [2] quantitatively reveals why discretization of the MHD equations is fundamentally more demanding than either the Maxwell or hydrodynamic equations alone. Unlike standard hydrodynamics, the energy analysis for MHD reveals the subtle role of the solenoidal condition in obtaining global and elementwise local stability through

strong or weak satisfaction of div B = 0 in element interiors
strong or weak satisfaction of [B·n] = 0 on element interfaces.

One immediate result from this theory are sufficient conditions to be imposed on numerical fluxes in hydrodynamics and magnetohydrodynamics so that energy stability and discrete cell entropy inequalities are rigorously obtained. The theory suggests several hybrid FEM discretizations strategies for compressible MHD depending on whether each condition is strongly or weakly enforced. In the remainder of the seminar, implementations of these concepts are explored in the context of both primal numerical (CFD) simulations as well as dual simulations used in error quantification and control using the theory developed by Becker and Rannacher [3,4].

[1] W. Reed and T. Hill, "Triangular mesh methods fo the neutron transport equation", Los Alamos Report, LA-UR-73-479, 1973.

[2] T. Barth, "On the Role of Involutions in Discontinuous Galerkin Discretization of Maxwell and MHD Systems", IMA Volumes in Mathematics and Applications, Springer-Verlag Pub., Vol. 142, 2005.

[3] R. Becker and R. Rannacher, "Weighted A-Posteriori Error Control in FE Methods," Proc. ENUMATH-97, Heidelberg, 1997.

[4] R. Becker and R. Rannacher, "An Optimal Control Approach to A-Posteriori Error Estimation in Finite Element Methods," Acta Numerica, Cambridge Press, 2001.

Last revised: 03/06/07 By: christov@math