Entropic sub-cell shock capturing schemes via Jin-Xin relaxation and Glimm front sampling for scalar hyperbolic conservation laws

Fr´ed´eric Coquel CNRS and Centre de Math´ematiques Appliqu´ees Shi Jin University of Wisconsin-Madison Jian-Guo Liu Duke University Li Wang University of California Los Angeles

Numerical Analysis and Scientific Computing mathscidoc:1702.25022

MATHEMATICS OF COMPUTATION, 2017.1
AbstractWe introduce a sub-cell shock capturing method for scalar conservation laws built upon the Jin-Xin relaxation framework. Here, sub-cell shock capturing is achieved thanks to an original defect measure correction technique. The proposed correction exactly restores entropy shock solutions of the exact Riemann problem and moreover, it produces monotone and entropy satisfying approximate self-similar solutions. These solutions are then sampled using Glimm's random choice method to advance in time. The resulting scheme combines the simplicity of the Jin-Xin relaxation method with the resolution of the Glimm's scheme in the sharp capture of discontinuities. The strong benefit of using defect measure corrections over usual sub-cell shock capturing methods is that the scheme can be easily made entropy satisfying with respect to infinitely many entropy pairs. Consequently, under a classical CFL condition, the method is proved to converge to the unique entropy weak solution of the Cauchy problem for general non-linear flux functions.
Conservation laws, Relaxation, Defect measure, Riemann problem, Shock capturing methods, Cell entropy condition, Glimm’s random choice
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@inproceedings{fr´ed´eric2017entropic,
  title={Entropic sub-cell shock capturing schemes via Jin-Xin relaxation and Glimm front sampling for scalar hyperbolic conservation laws},
  author={Fr´ed´eric Coquel, Shi Jin, Jian-Guo Liu, and Li Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170206154302927358193},
  booktitle={MATHEMATICS OF COMPUTATION},
  year={2017},
}
Fr´ed´eric Coquel, Shi Jin, Jian-Guo Liu, and Li Wang. Entropic sub-cell shock capturing schemes via Jin-Xin relaxation and Glimm front sampling for scalar hyperbolic conservation laws. 2017. In MATHEMATICS OF COMPUTATION. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170206154302927358193.
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