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A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems

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Abstract

In this article, a class of upwind schemes is proposed for systems, each of which yields an incomplete set of linearly independent eigenvectors. The theory of Jordan canonical forms is used to complete such sets through the addition of generalized eigenvectors. A modified Burgers’ system and its extensions generate δ,δ, δ,⋯,δn waves as solutions. The performance of flux difference splitting-based numerical schemes is examined by considering various numerical examples. Since the flux Jacobian matrix of pressureless gas dynamics system also produces an incomplete set of linearly independent eigenvectors, a similar framework is adopted to construct a numerical algorithm for a pressureless gas dynamics system.

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Acknowledgements

The author would like to thank Prof. G. D. Veerappa Gowda, Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore, and Prof. S. V. Raghurama Rao, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, for the fruitful discussions.

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Correspondence to Naveen Kumar Garg.

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Garg, N.K. A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems. Numer Algor 83, 1091–1121 (2020). https://doi.org/10.1007/s11075-019-00717-7

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Keywords

  • Weakly hyperbolic systems
  • Jordan canonical forms
  • Generalized eigenvectors
  • Upwind schemes

Mathematics Subject Classification (2010)

  • 15A21
  • 35L40
  • 35L65
  • 65M08