Compactness framework and convergence of Lax-Friedrichs and Godunov schemes for a nonstrictly hyperbolic system of conservation laws

Author:
Bruno Rubino

Journal:
Quart. Appl. Math. **53** (1995), 401-421

MSC:
Primary 35L65; Secondary 35D05, 65M06

DOI:
https://doi.org/10.1090/qam/1343459

MathSciNet review:
MR1343459

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Abstract | References | Similar Articles | Additional Information

Abstract: A compactness framework and a convergence theorem for the Lax-Friedrichs scheme and the Godunov scheme applied to the Cauchy problem for a nonstrictly hyperbolic system of conservation laws are established. The existence of weak solutions is proved using the theory of compensated compactness of Tartar, Murat, DiPerna, and Serre.

**[1]**G. Q. Chen, X. Ding, and P. Luo,*Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics*I, Acta Math. Sci.**5**, 415-432 (1985); II, Acta Math. Sci.**5**, 433-472 (1985)**[2]**G. Q. Chen,*Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics*III, Acta Math. Sci.**6**, 75-120 (1986)**[3]**K. N. Chueh, C. C. Conley, and J. A. Smoller,*Positively invariant regions for systems of non-linear diffusion equations*, Indiana Univ. Math. J.**26**, 372-411 (1977)**[4]**R. Courant and D. Hilbert,*Methods of mathematical physics*II:*Partial differential equations*, Wiley and Sons, 1962**[5]**M. G. Crandall and A. Majda,*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**, 1-21 (1980)**[6]**R. J. DiPerna,*Compensated compactness and general systems of conservation laws*, Trans. Amer. Math. Soc.**292**, 383-420 (1985)**[7]**R. J. DiPerna*Convergence of approximate solutions to conservation laws*, Arch. Rat. Mech. Anal.**82**, 27-70 (1983)**[8]**R. J. DiPerna,*Convergence of the viscosity method for isentropic gas dynamics*, Comm. Math. Phys.**91**, 1-30 (1983)**[9]**R. J. DiPerna,*Measure-valued solutions to conservation laws*, Arch. Rat. Mech. Anal.**88**, 223-270 (1985)**[10]**G. Glimm,*The interaction of non-linear hyperbolic waves*, Comm. Pure Appl. Math.**41**, 569-590 (1988)**[11]**S. K. Godunov,*Difference method for the numerical computation of discontinuous solutions of equations of hydrodynamics*, Mat. Sb.**47**, 271-306 (1959)**[12]**A. Harten,*High resolution schemes for hyperbolic conservation laws*, Jour. Comp. Phys.**49**, 357-393 (1983)**[13]**A. Harten, J. M. Hyman, and P. D. Lax,*On finite difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**, 297-322 (1976)**[14]**A. Harten, P. D. Lax, and B. van Leer,*On upstream differencing and Godunov-type schemes for hyperbolic conservation laws*, SIAM Review**25**, 35-61 (1983)**[15]**D. Hoff,*Invariant regions for systems of conservation laws*, Trans. Amer. Math. Soc.**289**, 591-610 (1985)**[16]**L. Hörmander,*Non-linear hyperbolic differential equations*, Lectures Notes 1986-1987, Lund University, Sweden, 1988 (mimeographed notes)**[17]**E. Isaacson, D. Marchesin, B. Plohr, and B. Temple,*The Riemann problem near a hyperbolic singularity: The classification of solutions of quadratic Riemann problems*I, SIAM Journal of Appl. Math.**48**, 1009-1032 (1988)**[18]**E. Isaacson and B. Temple,*The classification of solutions of quadratic Riemann problems*II, SIAM Journal of Appl. Math.**48**, 1287-1301 (1988); III, SIAM Journal of Appl. Math.**48**, 1302-1318 (1988)**[19]**P.-T. Kan,*On the Cauchy problem of a**system of non-strictly hyperbolic conservation laws*, Ph.D. thesis, Courant Institute of Math. Sciences, N.Y. University, 1989**[20]**B. L. Keyfitz and H. C. Kranzer,*A system of non-strictly hyperbolic conservation laws arising in elasticity theory*, Arch. Rat. Mech. Anal.**72**, 219-241 (1980)**[21]**S. Kružkov,*First order quasi-linear equations with several space variables*, Mat. Sb.**123**, 228-255 (1970)**[22]**P. D. Lax,*Weak solutions of nonlinear hyperbolic equations and their numerical computation*, Comm. Pure Appl. Math.**7**, 159-193 (1954)**[23]**P. D. Lax,*Hyperbolic systems of conservation laws*, II, Comm. Pure Appl. Math.**10**, 537-566 (1957)**[24]**P. D. Lax,*Shock waves and entropy*, in Contributions to Nonlinear Functional Analysis, ed. E. A. Zarantonello, Academic Press, 1971, pp. 603-634**[25]**P. D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, SIAM, Philadelphia, 1973**[26]**T. P. Liu,*Zero dissipative limit for conservation laws*, in Proceedings of the Conference on Non-linear Variational Problems and Partial Differential Equations, Isola d'Elba, 1990 (to appear)**[27]**P. Marcati,*Approximate solutions to the conservation laws via convective parabolic equations*, Comm. Partial Differential Equations**13**, 321-344 (1988)**[28]**P. Marcati and A. Milani,*The one-dimensional Darcy's law as the limit of a compressible Euler flow*, J. Differential Equations, 129-147 (1990)**[29]**F. Murat,*Compacité par compensation*, Ann. Scuola Normale Superiore Pisa**5**, 489-507 (1978)**[30]**I. G. Petrovskii,*Partial Differential Equations*, Iliffe Books, 1967**[31]**B. Rubino,*On the vanishing viscosity approximation to the Cauchy Problem for a**system of conservation laws*, Ann. Inst. H. Poincaré Anal. Non Linéaire**10**, 627-656 (1993)**[32]**D. G. Schaeffer and M. Shearer,*The classification of**systems of non-strictly hyperbolic conservation laws, with application to oil recovery*, Comm. Pure Appl. Math.**40**, 141-178 (1987)**[33]**D. G. Schaeffer and M. Shearer,*Riemann problems for non-strictly hyperbolic*2x2*systems of conservation laws*, Trans. Amer. Math. Soc.**304**, 267-306 (1987)**[34]**D. Serre,*La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations a une dimension d'espace*, J. Math. Pures Appl.**65**, 423-468 (1986)**[35]**J. A. Smoller,*Shock waves and reaction diffusion equations*, Springer-Verlag, 1983**[36]**S. L. Sobolev,*Partial Differential Equations of Mathematical Physics*, Pergamon Press, 1964**[37]**L. Tartar,*Compensated compactness and applications to partial differential equations*, in Nonlinear Analysis and Mechanics: Heriott-Watt Symposium, IV, Research Notes in Math., 1979, pp. 136-210**[38]**B. Temple,*Global existence of the Cauchy problem for a class of*2x2*non-strictly hyperbolic conservation laws*, Adv. Appl. Math.**3**, 355-375 (1982)

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DOI:
https://doi.org/10.1090/qam/1343459

Article copyright:
© Copyright 1995
American Mathematical Society