Kruzkov’s estimates for scalar conservation laws revisited

Bouchut, F.; Perthame, B.

doi:10.1090/S0002-9947-98-02204-1

Kruzkov’s estimates for scalar conservation laws revisited
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by F. Bouchut and B. Perthame

Trans. Amer. Math. Soc. 350 (1998), 2847-2870

DOI: https://doi.org/10.1090/S0002-9947-98-02204-1

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Abstract:

We give a synthetic statement of Kružkov-type estimates for multi-dimensional scalar conservation laws. We apply it to obtain various estimates for different approximation problems. In particular we recover for a model equation the rate of convergence in $h^{1/4}$ known for finite volume methods on unstructured grids.

Nous donnons un énoncé synthétique des estimations de type de Kružkov pour les lois de conservation scalaires multidimensionnelles. Nous l’appliquons pour obtenir d’estimations nombreuses pour problèmes différents d’approximation. En particulier, nous retrouvons pour une équation modèle la vitesse de convergence en $h^{1/4}$ connue pour les méthodes de volumes finis sur des maillages non structurés.

References

Philippe Bénilan and Ronald Gariepy, Strong solutions in $L^1$ of degenerate parabolic equations, J. Differential Equations 119 (1995), no. 2, 473–502. MR 1340548, DOI 10.1006/jdeq.1995.1099
F. Bouchut, Ch. Bourdarias, and B. Perthame, A MUSCL method satisfying all the numerical entropy inequalities, Math. Comp. 65 (1996), no. 216, 1439–1461. MR 1348038, DOI 10.1090/S0025-5718-96-00752-1
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), no. 2, 139–157. MR 1245008, DOI 10.1007/BF01385691
Bernardo Cockburn, Frédéric Coquel, and Philippe LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), no. 207, 77–103. MR 1240657, DOI 10.1090/S0025-5718-1994-1240657-4
B. Cockburn, F. Coquel, and P. G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), no. 3, 687–705. MR 1335651, DOI 10.1137/0732032
Bernardo Cockburn and Pierre-Alain Gremaud, A priori error estimates for numerical methods for scalar conservation laws. I. The general approach, Math. Comp. 65 (1996), no. 214, 533–573. MR 1333308, DOI 10.1090/S0025-5718-96-00701-6
Bernardo Cockburn and Pierre-Alain Gremaud, Error estimates for finite element methods for scalar conservation laws, SIAM J. Numer. Anal. 33 (1996), no. 2, 522–554. MR 1388487, DOI 10.1137/0733028
Constantine M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41. MR 303068, DOI 10.1016/0022-247X(72)90114-X
R. Eymard, T. Gallouët, R. Herbin, The finite volume method, to appear in Handbook of Numerical Analysis, Ph. Ciarlet and J.L. Lions, eds.
S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
N. N. Kuznecov, The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation, Ž. Vyčisl. Mat i Mat. Fiz. 16 (1976), no. 6, 1489–1502, 1627 (Russian). MR 483509
Şerban Gheorghiu, On some integrals, Gaz. Mat. Fiz. Ser. A 2 (1957), 70–75 (Romanian). MR 95255
Tai-Ping Liu and Michel Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), no. 3, 419–441. MR 735207, DOI 10.1016/0022-0396(84)90096-2
Bradley J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), no. 173, 59–69. MR 815831, DOI 10.1090/S0025-5718-1986-0815831-4
O. A. Oleĭnik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 165–170 (Russian). MR 0117408
B. Perthame, Convergence of N-schemes for linear advection equations, Trends in Applications of Mathematics to Mechanics, Pitman Mono. Ser. in Pure and Applied Math. 77, New-York (1995).
Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435, DOI 10.1090/S0025-5718-1983-0679435-6
Anders Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp. 53 (1989), no. 188, 527–545. MR 979941, DOI 10.1090/S0025-5718-1989-0979941-6
J.-P. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes, RAIRO Modél. Math. Anal. Numér. 28 (1994), no. 3, 267–295 (English, with English and French summaries). MR 1275345, DOI 10.1051/m2an/1994280302671
A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338