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Part of the book series: Applied Mathematical Sciences ((AMS,volume 118))

Abstract

We shall mainly consider in this section a two-dimensional p × p hyperbolic system

(1.1)

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Notes

  • Engquist, B. and A. Majda. Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1984), 629–651.

    Article  MathSciNet  Google Scholar 

  • Jeffrey, A. Quasilinear hyperbolic systems and waves, Research Notes in Mathematics 5, Pitman, London (1976).

    MATH  Google Scholar 

  • Hirsch, C. Numerical computation of internal and external flows, vol 2: Computational methods for inviscid and viscous flows, John Wiley Sons, Chichester, 1990, reprinted 1995.

    Google Scholar 

  • Richtmyer, R.D. and K.W. Morton. Difference Methods for Initial-Value Problems, Interscience, New York (1967).

    MATH  Google Scholar 

  • Chang, T. and L. Hsiao: The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics 41, Longman Scientific and Technical, Harlow, UK. 1989.

    Google Scholar 

  • Charrier, P., B. Dubroca, and L. Flandrin: Un solveur de Riemann approché pour l’étude d’écoulements hypersoniques bidimensionnels, C. R. Acad. Sci. Paris 317, Sér. I, (1993), 1083–1086.

    Google Scholar 

  • Charrier, P. and B. Tessieras. On front-tracking methods applied to hyperbolic systems of nonlinear conservation laws, SIAM J. Numer. Anal. 23 (1986), 461–472.

    Google Scholar 

  • Chen, G.-Q. The method of quasi-decoupling for discontinuous solutions of conservation laws, Arch. Rat. Mech. Anal. 121 (1992), 131–185.

    Article  MATH  Google Scholar 

  • Chen, G.-Q. and J. Glimm. Shock capturing and global solutions to the compressible Euler equations with geometrical structure, in Hyperbolic problems: theory, numerics, applications, Proceedings of the Fifth International Conference on Hyperbolic Problems, StonyBrook 1994, J. Glimm et al. (Eds.), World Scientific, Singapore (1996), 101–109.

    Google Scholar 

  • Liu, T.P. Solutions in the large for the equations of nonisentropic gas dynamics, Indiana Univ. Math. J. 26 (1977), 147–177.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, T.P. Admissible solutions of hyperbolic conservation laws, A.M.S. Memoirs 30 (240) (1981).

    Google Scholar 

  • Liu, T.-P. and Z. Xin. Overcompressive shock wave, in Nonlinear Evolution Equations that Change Type, IMA Volumes in Mathematics and its Applications 27, B.L. Keyfitz and M. Shearer (Eds.), Springer-Verlag, New York (1991), 139–145.

    Google Scholar 

  • Liu, T.-P. and L.-A. Ying. Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal. 26 (3) (1995), 519–528.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, X.B. A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws, SIAM J. Numer. Anal. 30 (1993), 701–716.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu X.D., S. Osher, and T. Chan. Weighted essentially non-oscillatory schemes, J. Comp. Phys. 115 (1994), 200–212.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, Y. and M. Vinokur. Nonequilibrium flow computations. An analysis of numerical formulations of conservation laws, J. Comp. Phys. 83 (1989), 373–397.

    Google Scholar 

  • Loh, C.Y. and W.H. Hui. A new Lagrangian method for steady supersonic flow computation I. Godunov scheme, J. Comp. Phys. 89 (1990), 207–240.

    Article  MathSciNet  MATH  Google Scholar 

  • Cargo, P. and G. Gallice: Un solveur de Roe pour les équations de la magnétohydrodynamique, C. R. Acad. Sci. Paris 320, Série I (1995), 1269–1272.

    Google Scholar 

  • Kreiss, H.-O. Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714.

    Google Scholar 

  • Kröner, D. Absorbing boundary conditions for the linearized Euler equations in 2-D, Math. Comp. 57 (195) (1991), 153–167.

    Article  MathSciNet  MATH  Google Scholar 

  • Kröner, D. and M. Rokyta. Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions, SIAM J. Numer. Anal. 31, 2 (1994), 324–343.

    Article  MathSciNet  MATH  Google Scholar 

  • Kruzkov, S. First-order quasilinear equations in several independent variables, Mat. Sb. 123 (1970), 228–255

    Google Scholar 

  • Kruzkov, S. English translation in Math. USSR. Sb. 10 (1970), 217–243.

    Article  Google Scholar 

  • Kumbaro, A. Modélisation, analyse mathématique et numérique des modèles bi-fluides d’écoulement diphasique, Doctoral dissertation, Thesis Université Paris-Sud, Orsay, France (1992).

    Google Scholar 

  • Kunhardt E.E. and C. Wu. Towards a more accurate Flux Corrected Transport algorithm, J. Comp. Phys. 68 (1987), 127–150.

    Article  MATH  Google Scholar 

  • Lallemand, M.H. Dissipative properties of Runge—Kutta schemes with upwind spatial approximation for the Euler equations, INRIA Research Report 1173 (1990), INRIA Rocquencourt, 78153 Le Chesnay, France.

    Google Scholar 

  • Lallemand, M.H., F. Fezoui, and E. Perez. Un schéma multigrille en éléments finis décentré pour les équations d’Euler, INRIA Research Report 602 (1987), INRIA Rocquencourt, 78153 Le Chesnay, France.

    Google Scholar 

  • Larrouturou, B. Recent progress in reactive flow computations, in Computing methods in applied sciences and engineering (Proceedings of the ninth conference on computing methods in applied sciences and engineering, INRIA, Paris 1990), R. Glowinski and A. Lichnewsky (Eds.), SIAM, Philadelphia (1990), 249–272.

    Google Scholar 

  • Larrouturou, B. How to preserve the mass fractions positivitiy when computing compressible multicomponent flows, J. Comp. Phys. 92 (1991), 273–295.

    Article  MathSciNet  Google Scholar 

  • Larrouturou, B. On upwind approximations of multi-dimensional multi-species flows, in Proceedings of the first European CFD Conference, Ch. Hirsh, J. Pénaux and E. Onate (Eds.), Elsevier, Amsterdam (1992), 117–126.

    Google Scholar 

  • Garcia-Navarro, P., M.E. Hubbard, and A. Priestley. Genuinely multidimensional upwinding for the 2d shallow water equations, J. Comp. Phys. 121 (1993), 79–93.

    Article  MathSciNet  Google Scholar 

  • Glaister, P. An approximate linearized Riemann solver for the Euler equations for real gases, J. Comp. Phys. 74 (1994), 382–408.

    Article  Google Scholar 

  • Blunt, M. and B. Rubin: Implicit flux limiting schemes for petroleum reservoir simulation, J. Comp. Phys. 102 (1992), 194–210.

    Article  MathSciNet  MATH  Google Scholar 

  • Raviart, P.A. and L. Sainsaulieu. Mathematical and numerical modelling of two-phase flows, in Computing Methods in Applied Sciences and Engineering, Proceedings of the tenth International Conference on Computing Methods in Applied Sciences and Engineering, France (1995), R. Glowinski (Ed.), Nova Science Publishers, Inc., New York, 119–132.

    Google Scholar 

  • Durlofsky, L., B. Engquist, and S. Osher. Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. Comp. Phys. 98 (1993), 64–73.

    Article  Google Scholar 

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Godlewski, E., Raviart, PA. (1996). The case of multidimensional systems. In: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol 118. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0713-9_5

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  • DOI: https://doi.org/10.1007/978-1-4612-0713-9_5

  • Publisher Name: Springer, New York, NY

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