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A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows

  • Chapter
Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

Part of the book series: Notes on Numerical Fluid Mechanics and Multidisciplinary Design ((NNFM,volume 120))

Abstract

In this paper we present a kinetic relaxation scheme for the Euler equations of gas dynamics in one space dimension. The method is easily applicable to solve any complex system of conservation laws. The numerical scheme is based on a relaxation approximation for conservation laws viewed as a discrete velocity model of the Boltzmann equation of kinetic theory. The discrete kinetic equation is solved by a splitting method consisting of a convection phase and a collision phase. The convection phase involves only the solution of linear transport equations and the collision phase instantaneously relaxes the distribution function to an equilibrium distribution. We prove that the first order accurate method is conservative, preserves the positivity of mass density and pressure and entropy stable. An anti-diffusive Chapman-Enskog distribution is used to derive a second order accurate method. The results of numerical experiments on some benchmark problems confirm the efficiency and robustness of the proposed scheme.

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Authors and Affiliations

  1. Institut für Geometrie und Praktische Mathematik, RWTH-Aachen, Templergraben-55, D-52056, Aachen, Germany

    K. R. Arun

  2. Institut für Mathematik, Johannes Gutenberg Universität Mainz, Staudingerweg 9, D-55099, Mainz, Germany

    M. Lukáčová-Medvidová

  3. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    Phoolan Prasad

  4. Department of Aerospace Engineering, Indian Institute of Science, Bangalore, 560012, India

    S. V. Raghurama Rao

Authors
  1. K. R. Arun
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  2. M. Lukáčová-Medvidová
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  3. Phoolan Prasad
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  4. S. V. Raghurama Rao
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Correspondence to K. R. Arun .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Rainer Ansorge

  2. , Department of Aerospace Engineering, Delft University of Technology, Delft, 2600, Netherlands

    Hester Bijl

  3. , Department of Mathematics and Natural Sc, University of Kassel, Heinrich-Plett-Str. 40, Kassel, 34132, Germany

    Andreas Meister

  4. Institute for Computational Mathematics, Technical University of Braunschweig, Pockelstr. 14, Braunschweig, 38106, Germany

    Thomas Sonar

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Arun, K.R., Lukáčová-Medvidová, M., Prasad, P., Raghurama Rao, S.V. (2013). A Second Order Accurate Kinetic Relaxation Scheme for Inviscid Compressible Flows. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33221-0_1

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  • DOI: https://doi.org/10.1007/978-3-642-33221-0_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33220-3

  • Online ISBN: 978-3-642-33221-0

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