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On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers

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Computing and Visualization in Science

Abstract

The paper is concerned with the numerical simulation of compressible flow with wide range of Mach numbers. We present a new technique which combines the discontinuous Galerkin space discretization, a semi-implicit time discretization and a special treatment of boundary conditions in inviscid convective terms. It is applicable to the solution of steady and unsteady compressible flow with high Mach numbers as well as low Mach number flow at incompressible limit without any modification of the Euler or Navier–Stokes equations.

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References

  1. Bassi F. and Rebay S. (1997). A high-order accurate finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131: 267–279

    Article  MATH  MathSciNet  Google Scholar 

  2. Bassi F. and Rebay S. (1997). High-order accurate discontinuous element solution of the 2D Euler equations. J. Comput. Phys. 138: 251–285

    Article  MATH  MathSciNet  Google Scholar 

  3. Baumann C.E. and Oden J.T. (1999). A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31: 79–95

    Article  MATH  MathSciNet  Google Scholar 

  4. Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.)(2000). Discontinuous Galerkin methods. Lecture Notes in Computational Science and Engineering, Vol. 11. Springer, Berlin Heidelberg New York

    Google Scholar 

  5. Dolejší V. (1998). Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1(3): 165–178

    Article  Google Scholar 

  6. Dolejší V. (2001). Anisotropic mesh adaptation technique for viscous flow simulation. East–West J. Numer. Math. 9(1): 1–24

    MathSciNet  Google Scholar 

  7. Dolejší V. (2004). On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations. Int. J. Numer. Methods Fluids 45: 1083–1106

    Article  Google Scholar 

  8. Dolejší, V., Feistauer, M.: On the discontinuous Galerkin method for the numerical solution of compressible high-speed flow. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, A. (eds.), Numerical Mathematics and Advanced Applications, ENUMATH 2001, pp. 65–84. Springer, Italia (2003)

  9. Dolejší, V., Feistauer, M., Hozman, J.: Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes. Comput. Methods Appl. Mech. Eng. (in press) (2007). Preprint No. MATH-knm-2005/4, Charles University Prague, School of Mathematics (2005)

  10. Dolejší V., Feistauer M. and Schwab C. (2003). On some aspects of the discontinuous Galerkin finite element method for conservation laws. Math. Comput. Simul. 61: 333–346

    Article  Google Scholar 

  11. Dolejší V., Feistauer M. and Sobotíková V. (2005). A Galerkin method for nonlinear convection–diffusion problems. Comput. Methods Appl. Mech. Eng. 194: 2709–2733

    Article  Google Scholar 

  12. Feistauer M. (1993). Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical, Harlow

    MATH  Google Scholar 

  13. Feistauer M., Felcman J. and Straškraba I. (2003). Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford

    MATH  Google Scholar 

  14. Feistauer M. and Švadlenka K. (2004). Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 12: 97–118

    Article  MATH  MathSciNet  Google Scholar 

  15. Feistauer, M., Švadlenka, K.: Space-time discontinuous method for solving nonstationary convection–reaction problems. Appl. Math. (in press) (2007)

  16. Klein R. (1995). Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics 1: one-dimensional flow. J. Comput. Phys. 121: 213–237

    Article  MATH  MathSciNet  Google Scholar 

  17. Lomtev I., Quillen C.W. and Karniadakis G.E. (1998). Spectral/hp methods for viscous compressible flows on unstructured 2D meshes. J. Comput. Phys. 144: 325–357

    Article  MATH  MathSciNet  Google Scholar 

  18. Meister, A., Struckmeier, J.: Hyperbolic Partial Differential Equations, Theory, Numerics and Applications. Vieweg, /Wiesbaden (2002)

  19. Roller S., Munz C.D., Geratz K. and Klein R. (1997). The multiple pressure variables method for weakly compressible fluids. Z. Angew. Math. Mech. 77: 481–484

    Google Scholar 

  20. Vijayasundaram G. (1986). Transonic flow simulation using upstream centered scheme of Godunov type in finite elements. J. Comput. Phys. 63: 416–433

    Article  MATH  MathSciNet  Google Scholar 

  21. Wesseling P (2001). Principles of Computational Fluid Dynamics. Springer, Berlin Heidelberg New York

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Miloslav Feistauer, Vít Dolejší & Václav Kučera

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  1. Miloslav Feistauer
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  2. Vít Dolejší
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  3. Václav Kučera
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Correspondence to Miloslav Feistauer.

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Communicated by P. Frolkovic.

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Feistauer, M., Dolejší, V. & Kučera, V. On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers. Comput. Visual Sci. 10, 17–27 (2007). https://doi.org/10.1007/s00791-006-0051-8

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  • DOI: https://doi.org/10.1007/s00791-006-0051-8

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