Nous étudions une famille de schémas non linéaires pour l'approximation numérique de l'advection linéaire sur grille quelconque en dimension d'espace supérieure à un. Une preuve de convergence est proposée à partir d'une estimation de la variation longitudinale. Cette estimation est une généralisation multidimensionnelle discrète de l'estimation TVD discrète, bien connue en dimension un d'espace.
We study a family of non linear schemes for the numerical solution of linear advection on arbitrary grids in several space dimension. A proof of weak convergence of the family of schemes is given, based on a new Longitudinal Variation Diminishing (LVD) estimate. This estimate is a multidimensional equivalent to the well-known TVD estimate in one dimension. The proof uses a corollary of the Perron-Frobenius theorem applied to a generalized Harten formalism.
@article{M2AN_2001__35_6_1159_0,
author = {Despr\'es, Bruno and Lagouti\`ere, Fr\'ed\'eric},
title = {Generalized Harten formalism and longitudinal variation diminishing schemes for linear advection on arbitrary grids},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {35},
year = {2001},
pages = {1159-1183},
mrnumber = {1873521},
zbl = {1005.76063},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2001__35_6_1159_0}
}
Després, Bruno; Lagoutière, Frédéric. Generalized Harten formalism and longitudinal variation diminishing schemes for linear advection on arbitrary grids. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) pp. 1159-1183. http://gdmltest.u-ga.fr/item/M2AN_2001__35_6_1159_0/
[1] , and, An unsplit higher order Godunov method for scalar conservation laws in multiple dimensions. J. Comp. Phys. 17 (1992) 1-24. | Zbl 0684.65088
[2] , and, Schémas d'équilibre pour des lois de conservation scalaires avec des termes sources raides. Report No. 3891, INRIA, France (2000).
[3] , First and second order schemes for a hyperbolic equation: convergence and error estimate, in Finite Volume for Complex Applications Problems and Perspectives, Benkhaldoun and Vilsmeier, Eds., Hermes, Paris (1997) 137-144.
[4] , and, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139-157. | Zbl 0801.65089
[5] , The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058
[6] , On the continuity in BV of the -projection into finite element spaces. Math. Comp. 57 (1991) 551-561. | Zbl 0736.47006
[7] , and, An error estimate for finite volume multidimensional conservation laws. Tech. Report No. 285 CMAPX, École Polytechnique, France (1993).
[8] and, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws: the multidimensional case. Math. Comp. 54 (1990) 545-581. | Zbl 0695.65066
[9] , Multidimensional upwind methods for hyperbolic conservation laws. J. Comp. Phys. 87 (1990) 171-200. | Zbl 0694.65041
[10] and, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach. Math. Comp. 57 (1991) 169-210. | Zbl 0741.35036
[11] and, Analyse numérique et calcul numérique pour les sciences et les techniques. Masson, Paris (1984). | Zbl 0708.35001
[12] , and, Compact advection schemes on unstructured grids, in Computational Fluid Dynamics Lect. Ser. 1993-04, von Karman Institute, Rhode-Saint-Genèse, Belgium (1993).
[13] and, Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire. C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999) 939-944. | Zbl 0944.76053
[14] and, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. In preparation. | Zbl 0999.76091
[15] , Measure value solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223-270. | Zbl 0616.35055
[16] and, A non-parametrized entropy correction for Roe's approximate Riemann solver. Numer. Math. 73 (1996) 169-208. | Zbl 0861.65073
[17] , and, Finite volume methods. Tech. Report No. 97-19, LATP, UMR 6632, Marseille, France. To appear in Handbook of Numerical Analysis, P.G. Ciarlet and J.-L. Lions, Eds., Elsevier, Amsterdam. | MR 1804748 | Zbl 0981.65095
[18] , Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984). | MR 775682 | Zbl 0545.49018
[19] and, Numerical approximation of hyperbolic systems of conservation laws, in Applied Mathematical Sciences 118, Springer, New York (1996). | MR 1410987 | Zbl 0860.65075
[20] and, Hyperbolic systems of conservation laws, in Mathématiques & Applications 3-4, Ellipses, Paris (1991). | MR 1304494 | Zbl 0768.35059
[21] , On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21 (1984) 1-23. | Zbl 0547.65062
[22] , High resolution schemes for hyperbolic conservation laws. J. Comp. Phys. 49 (1983) 357-393. | Zbl 0565.65050
[23] ,, and, Some results on uniformly High Order Accurate Essentially Non-oscillatory Schemes, in Adv. Numer. Appl. Math., ICASE Report No. 86-18, J.C. South, Jr and M.Y. Hussaini, Eds. (1986) 383-436; J. Appl. Numer. Math. 2 (1986) 347-377. | Zbl 0627.65101
[24] , Generalized solutions of the Cauchy problem in the large for non linear equations of first order. Dokl. Akad. Nauk SSSR 187 (1970) 29-32. English translation in Soviet Math. Dokl. 10 (1969). | Zbl 0202.37701
[25] , Finite difference schemes for multidimensional first order quasi-linear equations in classes of discontinuous functions. Probl. Math. Phys. Vych. Mat., Nauka, Moscow (1977) 181-194. | Zbl 0409.35005
[26] , Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. Thesis, Université Pierre et Marie Curie, Paris (2000).
[27] , Numerical Methods for Conservation Laws. ETHZ Zürich, Birkhäuser, Basel (1992). | MR 1153252 | Zbl 0847.65053
[28] , High-resolution conservative algorithms for advection in incompressible flows. SIAM J. Numer. Anal. 33 (1996) 627-665. | Zbl 0852.76057
[29] , and, A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994) 169-191. | Zbl 0820.35094
[30] and, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. | Zbl 0637.65091
[31] , Generalized formulations of TVD Lax-Wendroff schemes. ICASE Report No. 84-53, ICASE, NASA Langley Research Center, Hampton, VA (1984).
[32] and, Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal. 29 (1992) 1542-1568. | Zbl 0765.65093
[33] , High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 21 (1984) 995-1011. | MR 760628 | Zbl 0565.65048
[34] , Convergence of a streamline diffusion finite element method for conservation law with boundary conditions. RAIRO Modél. Math. Anal. Numér. 25 (1991) 749-783. | Numdam | Zbl 0751.65061
[35] , Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin, Heidelberg, New York (1997). | MR 1474503 | Zbl 0801.76062
[36] , Towards the ultimate conservative difference scheme, V. J. Comput. Phys 32 (1979) 101-136.
[37] , Matrix Iterative Analysis. 2. Printing, in Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ (1963). | MR 158502 | Zbl 0133.08602