Compactness framework and convergence of Lax-Friedrichs and Godunov schemes for a $2\times 2$ 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
Full-text PDF Free Access
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 $2 \times 2$ 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.
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)
G. Q. Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics III, Acta Math. Sci. 6, 75–120 (1986)
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)
R. Courant and D. Hilbert, Methods of mathematical physics II: Partial differential equations, Wiley and Sons, 1962
M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34, 1–21 (1980)
R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292, 383–420 (1985)
R. J. DiPerna Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82, 27–70 (1983)
R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91, 1–30 (1983)
R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88, 223–270 (1985)
G. Glimm, The interaction of non-linear hyperbolic waves, Comm. Pure Appl. Math. 41, 569–590 (1988)
S. K. Godunov, Difference method for the numerical computation of discontinuous solutions of equations of hydrodynamics, Mat. Sb. 47, 271–306 (1959)
A. Harten, High resolution schemes for hyperbolic conservation laws, Jour. Comp. Phys. 49, 357–393 (1983)
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)
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)
D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc. 289, 591–610 (1985)
L. Hörmander, Non-linear hyperbolic differential equations, Lectures Notes 1986–1987, Lund University, Sweden, 1988 (mimeographed notes)
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)
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)
P.-T. Kan, On the Cauchy problem of a $2 \times 2$ system of non-strictly hyperbolic conservation laws, Ph.D. thesis, Courant Institute of Math. Sciences, N.Y. University, 1989
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)
S. Kružkov, First order quasi-linear equations with several space variables, Mat. Sb. 123, 228–255 (1970)
P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, 159–193 (1954)
P. D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10, 537–566 (1957)
P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E. A. Zarantonello, Academic Press, 1971, pp. 603–634
P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, Philadelphia, 1973
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)
P. Marcati, Approximate solutions to the conservation laws via convective parabolic equations, Comm. Partial Differential Equations 13, 321–344 (1988)
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)
F. Murat, Compacité par compensation, Ann. Scuola Normale Superiore Pisa 5, 489–507 (1978)
I. G. Petrovskii, Partial Differential Equations, Iliffe Books, 1967
B. Rubino, On the vanishing viscosity approximation to the Cauchy Problem for a $2 \times 2$ system of conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 627–656 (1993)
D. G. Schaeffer and M. Shearer, The classification of $2 \times 2$ systems of non-strictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Math. 40, 141–178 (1987)
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)
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)
J. A. Smoller, Shock waves and reaction diffusion equations, Springer-Verlag, 1983
S. L. Sobolev, Partial Differential Equations of Mathematical Physics, Pergamon Press, 1964
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
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)
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)
G. Q. Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics III, Acta Math. Sci. 6, 75–120 (1986)
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)
R. Courant and D. Hilbert, Methods of mathematical physics II: Partial differential equations, Wiley and Sons, 1962
M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34, 1–21 (1980)
R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292, 383–420 (1985)
R. J. DiPerna Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82, 27–70 (1983)
R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91, 1–30 (1983)
R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88, 223–270 (1985)
G. Glimm, The interaction of non-linear hyperbolic waves, Comm. Pure Appl. Math. 41, 569–590 (1988)
S. K. Godunov, Difference method for the numerical computation of discontinuous solutions of equations of hydrodynamics, Mat. Sb. 47, 271–306 (1959)
A. Harten, High resolution schemes for hyperbolic conservation laws, Jour. Comp. Phys. 49, 357–393 (1983)
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)
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)
D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc. 289, 591–610 (1985)
L. Hörmander, Non-linear hyperbolic differential equations, Lectures Notes 1986–1987, Lund University, Sweden, 1988 (mimeographed notes)
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)
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)
P.-T. Kan, On the Cauchy problem of a $2 \times 2$ system of non-strictly hyperbolic conservation laws, Ph.D. thesis, Courant Institute of Math. Sciences, N.Y. University, 1989
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)
S. Kružkov, First order quasi-linear equations with several space variables, Mat. Sb. 123, 228–255 (1970)
P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, 159–193 (1954)
P. D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10, 537–566 (1957)
P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, ed. E. A. Zarantonello, Academic Press, 1971, pp. 603–634
P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, Philadelphia, 1973
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)
P. Marcati, Approximate solutions to the conservation laws via convective parabolic equations, Comm. Partial Differential Equations 13, 321–344 (1988)
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)
F. Murat, Compacité par compensation, Ann. Scuola Normale Superiore Pisa 5, 489–507 (1978)
I. G. Petrovskii, Partial Differential Equations, Iliffe Books, 1967
B. Rubino, On the vanishing viscosity approximation to the Cauchy Problem for a $2 \times 2$ system of conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 627–656 (1993)
D. G. Schaeffer and M. Shearer, The classification of $2 \times 2$ systems of non-strictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Math. 40, 141–178 (1987)
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)
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)
J. A. Smoller, Shock waves and reaction diffusion equations, Springer-Verlag, 1983
S. L. Sobolev, Partial Differential Equations of Mathematical Physics, Pergamon Press, 1964
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
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)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35L65,
35D05,
65M06
Retrieve articles in all journals
with MSC:
35L65,
35D05,
65M06
Additional Information
Article copyright:
© Copyright 1995
American Mathematical Society
