Un formalisme pour les estimations de type Kružkov pour les lois de conservation scalaires
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1996-1997), Talk no. 18, 12 p.

On montre comment le formalisme introduit récemment par l’auteur et Benoît Perthame permet de justifier la plupart des estimations d’erreurs pour des solutions approchées d’une loi de conservation scalaire.

Bouchut, François 1

1 Université d’Orléans et CNRS, UMR 6628, Département de Mathématiques, BP 6759, 45067 Orléans cedex 2, France
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     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
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Bouchut, François. Un formalisme pour les estimations de type Kružkov pour les lois de conservation scalaires. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1996-1997), Talk no. 18, 12 p. http://archive.numdam.org/item/SEDP_1996-1997____A18_0/

[1] F. Bouchut, Ch. Bourdarias, B. Perthame, A MUSCL method satisfying all the numerical entropy inequalities, Math. Comp. 65 (1996), 1439-1461. | MR | Zbl

[2] F. Bouchut, B. Perthame, Kružkov’s estimates for scalar conservation laws revisited, à paraî tre dans Trans. of the A.M.S. | Zbl

[3] Y. Brenier, Résolution d’équations d’évolution quasilinéaires en dimension N d’espace à l’aide d’équations linéaires en dimension N+1, J. Diff. Eq. 50 (1983), 375-390. | Zbl

[4] S. Champier, T. Gallouët, R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh, Numer. Math. 66 (1993), 139-157. | MR | Zbl

[5] B. Cockburn, F. Coquel, P. Le Floch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), 77-103. | MR | Zbl

[6] B. Cockburn, P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. Part I : The general approach, Math. Comp. 65 (1996), 533-573. | MR | Zbl

[7] B. Cockburn, P.-A. Gremaud, Error estimates for finite element methods for scalar conservation laws, SIAM J. Numer. Anal. 33 (1996), 522-554. | MR | Zbl

[8] B. Cockburn, P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. Part II : flux-splitting monotone schemes on irregular cartesian grids, prépublication. | Zbl

[9] R. Eymard, T. Gallouët, R. Herbin, The finite volume method, book to appear, « Handbook of Numerical Analysis », Ph. Ciarlet and J.L. Lions eds. | MR | Zbl

[10] E. Godlewski, P.-A. Raviart, Hyperbolic systems of conservation laws, Coll. Math. et Appl., Ellipses, Paris (1991). | MR | Zbl

[11] D. Hoff, The sharp form of Oleinik’s entropy condition in several space variables, Trans. of the A.M.S. 276 (1983), 707-714. | Zbl

[12] G. Jiang, C.-W. Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comp. 62 (1994), 531-538. | MR | Zbl

[13] S.N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243. | Zbl

[14] N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comp. Math. and Math. Phys. 16 (1976), 105-119. | Zbl

[15] P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566. | MR | Zbl

[16] B.J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), 59-69. | MR | Zbl

[17] H. Nessyahu, E. Tadmor, T. Tassa, The convergence rate of Godunov type schemes, SIAM J. Num. Anal. 31 (1994), 1-16. | MR | Zbl

[18] O.A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95-172. | MR | Zbl

[19] O.A. Oleinik, On the uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation, A.M.S. Transl. (2) 33 (1963), 285-290. | Zbl

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

[21] B. Perthame, Convergence of N-schemes for linear advection equations, Trends in Applications of Mathematics to Mechanics, Pitman M SPAM77, New-York (1995). | MR | Zbl

[22] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91-106. | MR | Zbl

[23] D. Serre, Systèmes de lois de conservation I et II, Diderot ed., Paris (1996). | MR

[24] J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, N.Y. (1983). | MR | Zbl

[25] A. Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for scalar conservation laws in two space dimensions, Math. Comp. (1989), 527-545. | MR | Zbl

[26] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Num. Anal. 28 (1991), 891-906. | MR | Zbl

[27] T. Tang, Z.-H. Teng, The sharpness of Kuznetsov’s O(Δx) L 1 -error estimate for monotone difference schemes, Math. Comp. 64 (1995), 581-589. | Zbl

[28] J.-P. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws I. Explicite monotone schemes, Math. Modeling and Num. Anal. 28 (1994), 267-295. | Numdam | MR | Zbl

[29] A.I. Vol’pert, The spaces BV and quasilinear equations, Math. USSR-Sbornik 2 (1967), 225-267. | Zbl