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

Bruno Després; Frédéric Lagoutière

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 6, page 1159-1183
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Després, Bruno, and Lagoutière, Frédéric. "Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids." ESAIM: Mathematical Modelling and Numerical Analysis 35.6 (2010): 1159-1183. <http://eudml.org/doc/197495>.

@article{Després2010,
abstract = { 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. },
author = {Després, Bruno, Lagoutière, Frédéric},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {LVD estimate; Harten Formalism; linear advection; finite volume methods.; weak convergence; longitudinal variation diminishing estimate; finite volume method; Perron-Frobenius theorem; generalized Harten formalism},
language = {eng},
month = {3},
number = {6},
pages = {1159-1183},
publisher = {EDP Sciences},
title = {Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids},
url = {http://eudml.org/doc/197495},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Després, Bruno
AU - Lagoutière, Frédéric
TI - Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 6
SP - 1159
EP - 1183
AB - 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.
LA - eng
KW - LVD estimate; Harten Formalism; linear advection; finite volume methods.; weak convergence; longitudinal variation diminishing estimate; finite volume method; Perron-Frobenius theorem; generalized Harten formalism
UR - http://eudml.org/doc/197495
ER -

References

top
  1. J.B. Bell, C.N. Dawson and G.R. Shubin, An unsplit higher order Godunov method for scalar conservation laws in multiple dimensions. J. Comp. Phys.17 (1992) 1-24.  
  2. 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).  
  3. 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.  
  4. S. Champier, T. Gallouët and R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math.66 (1993) 139-157.  
  5. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).  
  6. B. Cockburn, On the continuity in BV of the L2-projection into finite element spaces. Math. Comp.57 (1991) 551-561.  
  7. 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).  
  8. B. Cockburn and C.W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws: the multidimensional case. Math. Comp.54 (1990) 545-581.  
  9. P. Collella, Multidimensional upwind methods for hyperbolic conservation laws. J. Comp. Phys.87 (1990) 171-200.  
  10. F. Coquel and P. Le Floch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach. Math. Comp.57 (1991) 169-210.  
  11. R. Dautray and J.-L. Lions, Analyse numérique et calcul numérique pour les sciences et les techniques. Masson, Paris (1984).  
  12. 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).  
  13. 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.  
  14. B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. In preparation.  
  15. R.J. DiPerna, Measure value solutions to conservation laws. Arch. Rational Mech. Anal.88 (1985) 223-270.  
  16. F. Dubois and G. Mehlman, A non-parametrized entropy correction for Roe's approximate Riemann solver. Numer. Math.73 (1996) 169-208.  
  17. 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.  
  18. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984).  
  19. E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, in Applied Mathematical Sciences118, Springer, New York (1996).  
  20. E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws, in Mathématiques & Applications3-4, Ellipses, Paris (1991).  
  21. H. Harten, On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal.21 (1984) 1-23.  
  22. H. Harten, High resolution schemes for hyperbolic conservation laws. J. Comp. Phys.49 (1983) 357-393.  
  23. 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.  
  24. S.N. Kruzkov, Generalized solutions of the Cauchy problem in the large for non linear equations of first order. Dokl. Akad. Nauk SSSR187 (1970) 29-32. English translation in Soviet Math. Dokl.10 (1969).  
  25. 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.  
  26. 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).  
  27. R.J. LeVeque, Numerical Methods for Conservation Laws. ETHZ Zürich, Birkhäuser, Basel (1992).  
  28. R.J. LeVeque, High-resolution conservative algorithms for advection in incompressible flows. SIAM J. Numer. Anal.33 (1996) 627-665.  
  29. 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.  
  30. S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp.50 (1988) 19-51.  
  31. P.L. Roe, Generalized formulations of TVD Lax-Wendroff schemes. ICASE Report No. 84-53, ICASE, NASA Langley Research Center, Hampton, VA (1984).  
  32. P.L. Roe and D. Sidilkover, Optimum positive linear schemes for advection in two and three dimensions. SIAM J. Numer. Anal.29 (1992) 1542-1568.  
  33. P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal.21 (1984) 995-1011.  
  34. A. Szepessy, Convergence of a streamline diffusion finite element method for conservation law with boundary conditions. RAIRO Modél. Math. Anal. Numér.25 (1991) 749-783.  
  35. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin, Heidelberg, New York (1997).  
  36. B. Van Leer, Towards the ultimate conservative difference scheme, V. J. Comput. Phys32 (1979) 101-136.  
  37. R.S. Varga, Matrix Iterative Analysis. 2. Printing, in Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ (1963).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.