Generalized Harten formalism and longitudinal variation diminishing schemes for linear advection on arbitrary grids

Després, Bruno; Lagoutière, Frédéric

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 1159-1183.

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.

[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. | Zbl

[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. | Zbl

[5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR | Zbl

[6] B. Cockburn, On the continuity in BV of the $L^2$-projection into finite element spaces. Math. Comp. 57 (1991) 551-561. | Zbl

[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. | Zbl

[9] P. Collella, Multidimensional upwind methods for hyperbolic conservation laws. J. Comp. Phys. 87 (1990) 171-200. | Zbl

[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. | Zbl

[11] R. Dautray and J.-L. Lions, Analyse numérique et calcul numérique pour les sciences et les techniques. Masson, Paris (1984). | Zbl

[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. | Zbl

[14] B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics. In preparation. | Zbl

[15] R.J. Diperna, Measure value solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223-270. | Zbl

[16] F. Dubois and G. Mehlman, A non-parametrized entropy correction for Roe's approximate Riemann solver. Numer. Math. 73 (1996) 169-208. | Zbl

[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. | MR | Zbl

[18] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984). | MR | Zbl

[19] E. Godlewski and P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, in Applied Mathematical Sciences 118, Springer, New York (1996). | MR | Zbl

[20] E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws, in Mathématiques & Applications 3-4, Ellipses, Paris (1991). | MR | Zbl

[21] H. Harten, On a class of high resolution total-variation-stable finite-difference schemes. SIAM J. Numer. Anal. 21 (1984) 1-23. | Zbl

[22] H. Harten, High resolution schemes for hyperbolic conservation laws. J. Comp. Phys. 49 (1983) 357-393. | Zbl

[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. | Zbl

[24] S.N. Kruzkov, 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

[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. | Zbl

[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). | MR | Zbl

[28] R.J. Leveque, High-resolution conservative algorithms for advection in incompressible flows. SIAM J. Numer. Anal. 33 (1996) 627-665. | Zbl

[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. | Zbl

[30] S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. | Zbl

[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. | Zbl

[33] P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 21 (1984) 995-1011. | MR | Zbl

[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. | Numdam | Zbl

[35] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin, Heidelberg, New York (1997). | MR | Zbl

[36] B. Van Leer, Towards the ultimate conservative difference scheme, V. J. Comput. Phys 32 (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). | MR | Zbl