Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have beenwidely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (AnnMath 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable Communicated by Wolfgang Dahmen. B Eitan Tadmor [email protected] Ulrik S. Fjordholm [email protected] Roger Käppeli [email protected] Siddhartha Mishra [email protected] 1 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway 2 Seminar for Applied Mathematics, ETH Zürich, HG J 48, Rämistrasse 101, Zurich, Switzerland 3 Center for Scientific Computation and Mathematical Modeling (CSCAMM), Department of Mathematics, Institute for Physical Science and Technology (IPST), University of Maryland, College Park, MD 20742-4015, USA