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Residual Distribution Schemes on Quadrilateral Meshes

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Abstract

We propose an investigation of the residual distribution schemes for the numerical approximation of two-dimensional hyperbolic systems of conservation laws on general quadrilateral meshes. In comparison to the use of triangular cells, usual basic features are recovered, an extension of the upwinding concept is given, and a Lax–Wendroff type theorem is adapted for consistency. We show how to retrieve many variants of standard first and second-order accurate schemes. They are proven to satisfy this theorem. An important part of this paper is devoted to the validation of these schemes by various numerical tests for scalar equations and the Euler equations system for compressible fluid dynamics on non Cartesian grids. In particular, second-order accuracy is reached by an adaptation of the Linearity preserving property to quadrangle meshes. We discuss several choices as well as the convergence of iterative method to steady state. We also provide examples of schemes that are not constructed from an upwinding principle

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References

  1. Abgrall R. (2001). Toward the ultimate conservative scheme: following the quest. J. Comput. Phys. 167:277–315

    Article  MathSciNet  Google Scholar 

  2. Deconinck, H., Sermeus, K., and Abgrall, R. (2000). Status of multidimensional upwind residual distribution schemes and applications in aeronautics. AIAA Paper 2000-2328J.

  3. Deconinck, H., Struijs, R., Bourgeois, G., and Roe, P. L. (1993). Compact advection schemes on unstructured meshes. Computational Fluid Dynamics. VKI Lecture series 1993-04.

  4. Paillère, H., and Deconinck, H. (1997). Euler and Navier–Stokes Solvers using Multi-dimensional Upwind Schemes and Multigrid Acceleration, chapter Compact cell vertex convection schemes on unstructured meshes, Vieweg, Braunschweig, pp. 1–49

  5. van der Weide, E., and Deconinck, H. (1996). Positive matrix distribution schemes for hyperbolic systems, with application to the Euler equations. In Computational Fluid Dynamics, 3rd ECCOMAS CFD Conference, Wiley, New York.

  6. Abgrall R., and Mezine M. (2003). Construction of second-order accurate monotone and stable residual distribution schemes for unsteady problems. J. Comput. Phys. 188(1):16–55

    Article  MathSciNet  Google Scholar 

  7. Csík A., and Deconinck H. (2002). Space time residual distribution schemes for hyperbolic conservation laws on unstructured linear finite elements. In: Baines M.J (eds). Numerical Methods for Fluid Dynamics VII, Oxford, pp. 557–564.

  8. van der Weide, E., and Deconinck, H. (1995). Fluctuation splitting schemes for the Euler equations on quadrilateral grids. In Numerical methods for fluid dynamics V, Oxford, UK

  9. De Palma, P., Pascazio, G., Rubino, D. T., and Napolitano, M. (2004). Multidimensional Upwind Cell-vertex Schemes for Quadrilaterals, ECCOMAS CFD Conference 2004 Jyväskyl

  10. Chou, C.-S., and Shu, C.-W. (2006). High order Residual Distribution conservative finite difference schemes for steady states problems on non smooth meshes. J. Comput. Phys. (in press)

  11. Abgrall, R., Mer, K., and Nkonga, B. (2002). A Lax–Wendroff type theorem for residual schemes. In M. Hafez and J.J Chattot, editors, Innovative methods for numerical solutions of partial differential equations, World Scientific, Singapore pp. 243–266

  12. Csík Á., Ricchiuto M., and Deconinck H. (2003). A conservative formulation of the multidimensional upwind residual distribution schemes for general conservation laws. J. Comput. Phys. 179(1):286–312

    Article  Google Scholar 

  13. Abgrall R., and Roe P.L. (2003). Construction of very high order fluctuation scheme. J. Sci. Comput 19(1–3):3–36

    Article  MathSciNet  Google Scholar 

  14. Deconinck H., Roe P.L., and Struijs R. (1993). A multidimensional generalisation of Roe’s difference splitter for the Euler equations. Comput. Fluids 22:215–222

    Article  MathSciNet  Google Scholar 

  15. Paillère, H. (1995). Multi-dimensional Upwind Residual Distribution Schemes for the Euler and Navier-Stokes Equations on Unstructures Grids, Ph.D. thesis, Université Libre de Bruxelles

  16. Struijs, R., Deconinck, H., and Roe, P. L. (1991). Fluctuation splitting schemes for the 2D Euler equations. Computational luid Dynamics. VKI Lecture series 1991-01

  17. Abgrall R., and Mezine M. (2004). Construction of second-order accurate monotone and stable residual distribution schemes for steady problems. J. Comput. Phys. 195(2):474–507

    Article  MathSciNet  Google Scholar 

  18. Mezine, M. (2002). Conception de schémas distributifs pour l’aérodynamique stationnaire et instationnaire. Ph.D. thesis, Université Bordeaux 1

  19. Roe P.L., and Sidilkover D. (1992). Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal. 29(6):15–42

    Article  MathSciNet  Google Scholar 

  20. Godlewski, E., and Raviart, P. A. (1995). Numerical Approximation of Hyperbolic Systems of Conservation Laws. Volume 118 of Applied Mathematical Sciences, Springer. Berlin

  21. Abgrall, R. (2006). Essentially non oscillatory residual distribution schemes for hyperbolic problems. J. Comput. Phys. (in press)

  22. Ciarlet P.G. (1978). The Finite Element Method for Elliptic Problems. North Holland Publishing Company, Amsterdam

    MATH  Google Scholar 

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Authors and Affiliations

  1. Mathématiques Appliquées de Bordeaux, Université Bordeaux 1, 351 Cours de la Libération, 33405, Talence Cedex, France

    R. Abgrall & F. Marpeau

  2. Institut Universitaire de France and Projet Scalapplix, INRIA FutURs, Paris, France

    R. Abgrall

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  1. R. Abgrall
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  2. F. Marpeau
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Correspondence to R. Abgrall.

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Abgrall, R., Marpeau, F. Residual Distribution Schemes on Quadrilateral Meshes. J Sci Comput 30, 131–175 (2007). https://doi.org/10.1007/s10915-005-9023-2

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  • DOI: https://doi.org/10.1007/s10915-005-9023-2

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