Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T04:21:30.700Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids

Published online by Cambridge University Press:  15 April 2002

Bruno Després  and
Frédéric Lagoutière
Bruno Després
Affiliation:
Commissariat à l'Énergie Atomique, BP 12, 91680 Bruyères-le-Châtel, France. ([email protected]) Laboratoire d'analyse numérique, Université de Paris VI, 175 rue du Chevaleret, 75013 Paris, France. ([email protected])
Frédéric Lagoutière
Affiliation:
Laboratoire d'analyse numérique, Université de Paris VI, 175 rue du Chevaleret, 75013 Paris, France. ([email protected])
Get access

Abstract

We study a family of non linear schemes for the numerical solution oflinear 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.

Keywords

LVD estimateHarten Formalismlinear advectionfinite volume methods.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 6 , November 2001 , pp. 1159 - 1183
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bell, J.B., Dawson, C.N. and Shubin, G.R., An unsplit higher order Godunov method for scalar conservation laws in multiple dimensions. J. Comp. Phys. 17 (1992) 1-24.
R. Botchorishvili, B. Perthame and A. Vasseur, Schémas d'équilibre pour des lois de conservation scalaires avec des termes sources raides. Report No. 3891, INRIA, France (2000).
C. Chainais-Hillairet, 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.
Champier, S., Gallouët, T. and Herbin, R., Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139-157. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
Cockburn, B., On the continuity in BV of the L 2-projection into finite element spaces. Math. Comp. 57 (1991) 551-561.
B. Cockburn, F. Coquel and P. Le Floch, An error estimate for finite volume multidimensional conservation laws. Tech. Report No. 285 CMAPX, École Polytechnique, France (1993).
Cockburn, B. and Shu, C.W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws: the multidimensional case. Math. Comp. 54 (1990) 545-581.
Collella, P., Multidimensional upwind methods for hyperbolic conservation laws. J. Comp. Phys. 87 (1990) 171-200. CrossRef
Coquel, F. and Le Floch, P., Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach. Math. Comp. 57 (1991) 169-210. CrossRef
R. Dautray and J.-L. Lions, Analyse numérique et calcul numérique pour les sciences et les techniques. Masson, Paris (1984).
H. Deconinck, R. Struijs and G. Bourgeois, Compact advection schemes on unstructured grids, in Computational Fluid Dynamics Lect. Ser. 1993-04, von Karman Institute, Rhode-Saint-Genèse, Belgium (1993).
B. Després and F. Lagoutière, Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire. C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999) 939-944.
B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. In preparation.
DiPerna, R.J., Measure value solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223-270. CrossRef
Dubois, F. and Mehlman, G., A non-parametrized entropy correction for Roe's approximate Riemann solver. Numer. Math. 73 (1996) 169-208. CrossRef
R. Eymard, T. Gallouët and R. Herbin, 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.
E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984).
E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, in Applied Mathematical Sciences 118, Springer, New York (1996).
E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws, in Mathématiques & Applications 3-4, Ellipses, Paris (1991).
Harten, H., On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21 (1984) 1-23. CrossRef
Harten, H., High resolution schemes for hyperbolic conservation laws. J. Comp. Phys. 49 (1983) 357-393. CrossRef
A. Harten, S. Osher, B. Engquist and S. Chakravarthy, 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.
Kruzkov, S.N., 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).
N.N. Kuznetzov, 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.
F. Lagoutière, Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. Thesis, Université Pierre et Marie Curie, Paris (2000).
R.J. LeVeque, Numerical Methods for Conservation Laws. ETHZ Zürich, Birkhäuser, Basel (1992).
LeVeque, R.J., High-resolution conservative algorithms for advection in incompressible flows. SIAM J. Numer. Anal. 33 (1996) 627-665. CrossRef
P.-L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994) 169-191.
Osher, S. and Tadmor, E., On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. CrossRef
P.L. Roe, Generalized formulations of TVD Lax-Wendroff schemes. ICASE Report No. 84-53, ICASE, NASA Langley Research Center, Hampton, VA (1984).
Roe, P.L. and Sidilkover, D., Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal. 29 (1992) 1542-1568. CrossRef
Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 21 (1984) 995-1011. CrossRef
Szepessy, A., Convergence of a streamline diffusion finite element method for conservation law with boundary conditions. RAIRO Modél. Math. Anal. Numér. 25 (1991) 749-783. CrossRef
E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin, Heidelberg, New York (1997).
Van Leer, B., Towards the ultimate conservative difference scheme, V. J. Comput. Phys 32 (1979) 101-136. CrossRef
R.S. Varga, Matrix Iterative Analysis. 2. Printing, in Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ (1963).