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Multigrid Methods III pp 253–263Cite as

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Multigrid Methods for Hyperbolic Equations

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Abstract

A multigrid method for solving hyperbolic partial differential equations is presented. A second order conservative discretisation of the differential equation is obtained by a cell vertex scheme which has been introduced by Ni [1]. For the smoothing iteration a truncated Lax-Wendroff scheme is applied. A frequency decomposition approach uses multiple coarse grid corrections and avoids the need for numerical dissipation. The coarse grid system is a modified Galerkin product. Numerical results for a two-grid iteration for a linear periodic initial value problem are presented. Convergence rates of 0.5 are observed. Convergence deteriorates when the characteristics are aligned with the grid.

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References

  1. Ni, Ron-Ho: A multiple-grid scheme for solving the Euler equations. AIAA Paper 81-1025, 1981, or AIAA J. 20 (1982), pp. 1565–1571.

    Google Scholar 

  2. Brandt, A.; Dinar, N.: Multigrid solutions to elliptic flow problems. in: Parter, S. V. (ed.): Numerical methods for partial differential equations. New York, N. Y.: Academic Press, 1979, pp. 53–147.

    Google Scholar 

  3. Hackbusch, W. Trottenberg, U. (eds.): Multigrid methods. Köln, 1981. Proceedings: (Lecture Notes in Mathematics, 960.) Berlin, Heidelberg, New York: Springer, 1982.

    Google Scholar 

  4. Hackbusch, W.: Multi-grid methods and applications. Berlin, Heidelberg, New York: Springer, 1985.

    Book  Google Scholar 

  5. Kroll, N.: Einsatz von Mehrgittertechniken zur Berechnung stationärer reibungsloser Strömungen auf der Basis der Eulerschen Bewegungsgleichungen — Literaturstudie. (The use of multigrid methods based on the Euler equations for calculating steady inviscid flows. — A literature study.) Braunschweig, (Germany): Institut für Entwurfsaerodynamik; German Aerospace Research Center, internal Report: DFVLR-IB 129-85/9, 1985.

    Google Scholar 

  6. Hall, M. G.: Cell-vertex multigrid schemes for solution of the Euler equations. in: Morton, K., W.; Baines, M., J.; (eds.): Proc. Numerical Methods for Fluid Dynamics, II. Reading, 1985. Oxford: Clarendon Press, 1986, pp. 303–345.

    Google Scholar 

  7. Jameson, A.: Computational transonics. Comm. Pure Appl. Math. 41 (1988), pp. 507–549.

    Article  Google Scholar 

  8. Kroll, N.; Radespiel, R.; Rossow, C.-C: Experiences with explicit time-stepping schemes for supersonic flow fields. in: Wesseling, P.(ed.): Eighth GAMM-Conf. Numer. Methods in Fluid Mechanics. Delft, 1989. Proceedings: (Notes on Numer. Fluid Mechanics 29). Braunschweig: Vieweg, 1990, pp. 252–261.

    Google Scholar 

  9. Jameson, A.; Schmidt, W.; Turkel, E.: Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes. AIAA paper 81-1259, 1981.

    Google Scholar 

  10. Decker, N. H.; Turkel, E.: Multigrid for hypersonic inviscid flows. Third Int. Conf. on Hyperbolic Problems, 1990. ICASE Rep. No. 90-54.

    Google Scholar 

  11. Wittum, G.: Multi-grid methods for Stokes and Navier-Stokes equations. Transforming smoothers — algorithms and numerical results. Numerische Mathematik 54 (1989), pp. 543–563.

    Article  Google Scholar 

  12. Brandt, A.: General PDE systems at vanishing ellipticity and other projects. Third European Conf. on Multigrid Methods, Bonn, 1990.

    Google Scholar 

  13. Gustafsson, B.; Lötstedt, P.: Convergence of the multigrid method for systems of first order differential equations. Third European Conf. on Multigrid Methods, Bonn, 1990.

    Google Scholar 

  14. Hackbusch, W.: A new approach to robust multi-grid solvers. in: McKenna, J.; Temam, R. (ed.): ICI AM’ 87: First Int. Conf. on Industrial and Appl. Math. Paris, 1987. Proc: Philadelphia, PA: SIAM, 1988, pp. 114-126.

    Google Scholar 

  15. Hackbusch, W.: The frequency decomposition multi-grid algorithm. in: Hackbusch, W. (ed.): Fourth GAMM-Seminar Kiel, 1988: Robust multi-grid methods. Proc: (Notes on Numer. Fluid Mechanics 23). Braunschweig: Vieweg, 1988, pp. 96–104.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Informatics and Applied Mathematics, Christian-Albrechts University, D-2300, Kiel, Germany

    Edgar Katzer

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  1. Edgar Katzer
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Editors and Affiliations

  1. Institut für Informatik und Praktische Mathematik, Universität Kiel, Olshausenstrasse 40, D-2300, Kiel 1, Germany

    W. Hackbusch

  2. Gesellschaft für Mathematik und Datenverarbeitung mbH, Postfach 1240, Schloss Birlinghoven, D-5205, Sankt Augustin 1, Germany

    U. Trottenberg

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© 1991 Springer Basel AG

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Katzer, E. (1991). Multigrid Methods for Hyperbolic Equations. In: Hackbusch, W., Trottenberg, U. (eds) Multigrid Methods III. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 98. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5712-3_18

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  • DOI: https://doi.org/10.1007/978-3-0348-5712-3_18

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5714-7

  • Online ISBN: 978-3-0348-5712-3

  • eBook Packages: Springer Book Archive

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