Convergence of a finite volume extension of the Nessyahu-Tadmor scheme on unstructured grids for a two-dimensional linear hyperbolic equation

SIAM Journal on Numerical Analysis

Volumn 36, Issue 3, 1999, Pages 738-771

  Arminjon, Paul ^a   Viallon, Marie Claude ^b  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

^b UNIVERSITÉ JEAN MONNET   (France)

Author keywords

Barycentric cells; Double ellipse; Finite volumes; L infinity estimate; Least squares limiters; MUSCL; Staggered unstructured triangular grids; Weak convergence

Indexed keywords

EID: 0000954323     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142994278899     Document Type: Article

References (37)

1. F. ANGRAND, A. DERVIEUX, L. LOTH, AND G. VIJAYASUNDARAM, Simulations of Euler transonic flows by means of explicit finite-element type schemes, INRIA Res. Rep. 250, Le Chesnay, France, 1983.
2. F. ANGRAND AND A. DERVIEUX, Some explicit triangular finite element schemes for the Euler equations, Internat. J. Numer. Methods Fluids, 4 (1984), pp. 749-764.
3. P. ARMINJON, A. DERVIEUX, L. FEZOUI, H. STEVE, AND B. STOUFFLET, Non-oscillatory schemes for multidimensional Euler calculations with unstructured grids, in Proceedings 2nd International Conference on Hyperbolic Problems, Aachen, Germany, 1988, R. Jeltsch and J. Ballmann, eds., Notes Numer. Fluid Mech., 24, Vieweg, Braunschweig, 1988, pp. 1-10.
4. P. ARMINJON AND A. DERVIEUX, Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids, J. Comput. Phys., 106 (1993), pp. 176-198.
5. P. ARMINJON, Two-dimensional finite volume extension of the Nessyahu-Tadmor scheme for conservation laws, private communication to A. Jameson, Princeton University, Princeton, NJ, 1992.
6. P. ARMINJON AND M.C. VIALLON, Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d'espace, C. R. Acad. Sci., Paris, Sér. I Math., 320 (1995), pp. 85-88.
7. P. ARMINJON, M.C. VIALLON, AND A. MADRANE, A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservationlaws on unstructured grids, revised version with numerical applications, Internat. J. Comp. Fluid Dynamics, 9 (1997), pp. 1-22.
8. P. ARMINJON AND A. DERVIEUX, Incremental non-oscillatory finite volume schemes for compressible flows, in Proceedings 1st International Conference Computational Fluid Dynamics Society of Canada, Montréal, 1993, R. Baliga and W.G. Habashi, eds., CFD Society of Canada, 1993, pp. 91-102.
9. P. ARMINJON, D. STANESCU, AND M.C. VIALLON, A two-dimensional finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flows, in A Collection of Technical Papers, Proceedings of the 6th International Symposium on Comp. Fluid Dynamics, Lake Tahoe, NV, M. Hafez, ed., University of California, Davis, CA, 4, pp. 7-14.
10. P. ARMINJON, M.C. VIALLON, AND A. MADRANE, (1995). From Lax-Friedrichs to a multidimensional finite volume extension of the Nessyahu-Tadmor scheme for compressible flows, in Proceedings International Conference on Numerical Methods for the Euler and Navier-Stokes Equations, Centre de Recherches Mathématiques, Université de Montréal, 1995, P. Arminjon and A. Dervieux, eds., final version of the Proceeding to be published in the American Math. Soc.-Centre de Recherches Mathématiques Conference Proceedings Series.
11. P. ARMINJON, A. MADRANE, AND M.C. VIALLON, Comparison of a finite volume version of the Lax-Friedrichs and Nessyahu-Tadmor schemes and a discontinuous finite element method for compressible flows on unstructured grids, in Godunov's method for Gas Dynamics, Proceedings of symposium honoring S.K. Godunov, 1997, University of Michigan, Ann Arbor, MI, B. van Leer, ed., J. Comput. Phys., to appear.
12. P. ARMINJON AND A. MADRANE, A mixed finite volume/finite element method for 2-dimensional compressible Navier-Stokes equations on unstructured grids, in Proceedings 7th International Conference on Hyperbolic Problems, ETH Zürich, M. Fey and R. Jeltsch, eds., 1998.
13. P. ARMINJON AND A. MADRANE, A staggered Lax-Friedrichs type mixed finite volume/finite element method for the simulation of viscous compressible flows on unstructured triangular grids, AIAA Journal, to appear.
14. S. CHAMPIER AND T. GALLOUËT, Convergence d'un schéma décentré amont pour une équation hyperbolique linéaire sur un maillage triangulaire, RAIRO Modél. Math. Anal. Numer., 26 (1992), pp. 835-853.
15. B. COCKBURN, S. HOU, AND C.W. SHU, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: The multidimensional case, Math. Comp., 54 (1990), pp. 545-581.
16. B. COCKBURN, F. COQUEL, AND P. LE FLOCK, An error estimate for finite volume methods for multi-dimensional conservation laws, Math. Comp., 63 (1994), pp. 77-103.
17. M. GEIBEN, Convergence of MUSCL-Type Finite Volume Schemes on Unstructured Triangular Grids, SFB 256 Preprint 318, University of Bonn, Germany, 1993.
18. J. JAFFRÉ AND L. KADDOURI, Discontinuous finite elements for the Euler equations, in Proceedings 3rd International Conference on Hyperbolic Problems, Uppsala, Sweden, B. Engquist and B. Gustafsson, eds., Studentlitteratur, Lund, Sweden, 1991, pp. 602-610.
19. D. KRÖNER, S. NOELLE, AND M. ROKYTA, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions, SFB 256 Preprint 368, Univ. of Bonn, Germany; Numer. Math., 71 (1993), pp. 527-560.
20. D. KRÖNER AND M. ROKYTA, Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions, SIAM J. Numer. Anal., 31 (1994), pp. 324-343.
21. P.D. LAX AND B. WENDROFF, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), pp. 217-237.
22. B. VAN LEER, Towards the ultimate conservative difference scheme II: Monotonicity and conservation combined in a second order scheme, J. Comput. Phys., 14 (1974), pp. 361-370.
23. B. VAN LEER, Towards the ultimate conservative difference scheme IV: A new approach to numerical convection, J. Comput. Phys., 23 (1977), pp. 276-299.
24. X.D. LIU, A maximum principle satisfying modification of triangle based adaptive stencils for the solution of scalar hyperbolic conservation laws, SIAM J. Numer. Anal, 30 (1993), pp. 701-716.
25. H. NESSYAHU AND E. TADMOR, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Physics, 87 (1990), pp. 408-463.
26. S. NOELLE, Convergence of higher order finite volume schemes on irregular grids, Adv. Comput. Math., 3 (1995), pp. 197-218.
27. S. NOELLE, A note on entropy inequalities and error estimates for higher order accurate finite volume schemes on irregular families of grids, Math. Comp., 65 (1996), pp. 1155-1163.
28. J. PERAIRE, M. VAHDATI, K. MORGAN, AND O.C. ZIENKIEWICZ, Adaptive remeshing for compressible flow computations, J. Comput. Phys., 72 (1987), pp. 449-466.
29. R.D. RICHTMYER AND K.W. MORTON, Difference Methods for Initial Value Problems, 2nd ed., Interscience and John Wiley, New York, 1967.
30. P.L. ROE, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357-372.
31. M.C. VIALLON AND P. ARMINJON, Convergence du schéma de Nessyahu-Tadmor sur des maillages non-structurés pour une équation hyperbolique linéaire bidimensionnelle, Rapport de recherche 183, Equipe d'Analyse Numérique, Universités de Lyon-St-Êtienne, 1994.
32. M.C. VIALLON AND P. ARMINJON, Convergence of a finite volume extension of the Nessyahu-Tadmor Scheme on unstructured grids for a two-dimensional linear hyperbolic equation, Res. Report CRM-2239, Centre de Recherches Mathématiques, Univ. de Montréal, Montréal, Canada, revised version, 1996.
33. P. ARMINJON, M.C. VIALLON, A. MADRANE, AND H. KADDOURI, Discontinuous finite elements and a 2-dimensional finite volume generalization of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flows on unstructured grids, in Computational Fluid Dynamics Review 1998, Vol. 1, M. Hafez and K. Oshima, eds., World Scientific, 1998, pp. 241-261.
34. INRIA AND GAMNI-SMAI, Workshop on Hypersonic Flows for Reentry Problems, Problem 6: Flow over a double ellipse, test case 6.1: Non-Reactive Flows, Antibes, France, 1990.
35. M.J. CASTRO DIAZ AND F. HECHT, Anisotropic Surface Mesh Generation, INRIA Res. Rep. 2672, INRIA, Le Chesnay, France, 1995.
36. G.S. JIANG AND E. TADMOR, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, UCLA CAM Report 96-36, UCLA, Los Angeles, CA, 1996, SIAM J. Sci. Comput., 19 (1998), pp. 1892-1917.
37. D. LEVY AND E. TADMOR, Non-oscillatory central schemes for the incompressible 2-D Euler equations, Math Res. Lett., 4 (1997), pp. 1-20.