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Construction of approximate entropy measure valued solutions for systems of conservation laws.

by U. Fjordholm and R. Kappeli and S. Mishra and E. Tadmor

(Report number 2014-33)

Abstract
Numerical evidence is presented to demonstrate that state of the art numerical schemes need not converge to entropy solutions of systems of hyperbolic conservation laws in several space dimensions. Combined with recent results on the lack of stability of these solutions, we advocate the more general notion of entropy measure valued solutions as the appropriate paradigm for solutions of such multi-dimensional systems. We propose a detailed numerical procedure which constructs approximate entropy measure valued solutions, and we prove sufficient criteria that ensure their (narrow) convergence, thus providing a viable numerical framework for the approximation of entropy measure valued solutions. Examples of schemes satisfying these criteria are presented. A number of numerical experiments, illustrating our proposed procedure and examining interesting properties of the entropy measure valued solutions, are also provided.

Keywords: Hyperbolic conservation laws, uniqueness, stability, entropy condition, measure-valued solutions, atomic initial data, random field, weak BV estimate, narrow convergence

@Techreport{FKMT14_583,
  author = {U. Fjordholm and R. Kappeli and S. Mishra and E. Tadmor},
  title = {Construction of approximate entropy measure valued solutions for systems of conservation laws.},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-33},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-33.pdf },
  year = {2014}
}

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