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.
@article{SEDP_1996-1997____A18_0, author = {Bouchut, Fran\c{c}ois}, title = {Un formalisme pour les estimations de type {Kru\v{z}kov} pour les lois de conservation scalaires}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"}, note = {talk:18}, pages = {1--12}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {1996-1997}, zbl = {1069.35508}, mrnumber = {1482824}, language = {fr}, url = {http://www.numdam.org/item/SEDP_1996-1997____A18_0/} }
TY - JOUR AU - Bouchut, François TI - Un formalisme pour les estimations de type Kružkov pour les lois de conservation scalaires JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:18 PY - 1996-1997 SP - 1 EP - 12 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_1996-1997____A18_0/ LA - fr ID - SEDP_1996-1997____A18_0 ER -
%0 Journal Article %A Bouchut, François %T Un formalisme pour les estimations de type Kružkov pour les lois de conservation scalaires %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:18 %D 1996-1997 %P 1-12 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_1996-1997____A18_0/ %G fr %F SEDP_1996-1997____A18_0
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), Exposé no. 18, 12 p. http://www.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 , 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 -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