Graduate Seminar "Aktuelle Themen der Numerik"
Thursday, Jul 17, 2014, 02:00 pm
Analysis of a shock-capturing discontinuous Galerkin scheme for hyperbolic systems of conservation laws using measure-valued solutions
Mohammad Zakerzadeh M.Sc. (AICES, RWTH Aachen)
The design and analysis of numerical schemes for nonlinear systems of conservation laws is made difficult by the fact that there is no global well-posedness theory for most interesting problems. Regardless of the particular type of numerical scheme under investigation, the scope of the analysis often restricted to stability estimates. Stronger proofs of convergence are difficult to come by and often hinge on an appropriate notion of convergence and the concept of solution.
Entropy-measure-valued (emv) solutions  are a generalization of traditional entropy-weak solutions to conservation laws. It is a classic result for generic scalar conservation laws that emv solutions reduce to traditional entropy-weak solutions if the initial data is a Dirac measure . For multidimensional systems such general results are not available. Numerical evidence makes it doubtful that entropy-weak solutions constitute the appropriate solution paradigm for this case, and it has been conjectured that measure-valued solutions ought to be considered the appropriate notion of solution .
Building on previous results by Hiltebrand and Mishra , we prove that bounded solutions of a certain class of space-time Discontinuous Galerkin (DG) schemes converge to an emv solution. The novelty in our work is that no streamline-diffusion terms are used for stabilization. While streamline diffusion stabilization is often included in the analysis of DG schemes (see  and references cited therein), it is not commonly found in practical implementations. We show that a properly chosen nonlinear shock-capturing operator suffices to provide the necessary stability and entropy consistency estimates. Our results are valid for arbitrary degree of polynomial
 R. J. DiPerna, “Measure-valued solutions to conservation laws ”, Arch. Ration. Mech. Anal.,
88(3), 223–270, (1985).
 H. Lim, Y. Yu, J. Glimm, X. L. Li, D. H. Sharp, “Chaos, transport and mesh convergence
for fluid mixing”, Acta Math. Appl. Sin. , 24(3), 355–368 (2008).
 A. Hiltebrand and S. Mishra, “Entropy stable shock capturing space-time discontinuous
Galerkin schemes for systems of conservation laws ”,Numer. Math., 24(3), 103–151 (2014).
Time: 02:00 pm
Location: AICES seminar room R 115, Rogowski-Gebäude, Schinkelstr. 2, 52062 Aachen