Convergence of relaxation schemes to the equations of elastodynamics

Mathematics of Computation

Volumn 70, Issue 234, 2001, Pages 555-577

  Gosse, Laurent ^a   Tzavaras, Athanasios E ^b  

^a INSTITUTE OF COMPUTER SCIENCE   (Greece)

^b UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Compensated compactness; Relaxation schemes

Indexed keywords

EID: 0035531372     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-00-01256-4     Document Type: Article

References (23)

1. D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), 163-193. CMP 98:13
2. D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multi-dimensional conservation laws, (1998), (preprint).
3. H. Brezis, Analyse Fonctionnelle, Théorie et Applications Masson (1983). MR 85a:46001
4. G.-Q. Chen, C.D Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), 789-830. MR 95h:35133
5. F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal. 35 (1998), 2223-2249. MR 2000a:76129
6. R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Analysis 82 (1983), 27-70. MR 84k:35091
7. C. Faciu and M. Mihailescu-Suliciu, The energy in one-dimensional rate-type semilinear viscoelasticity, Int. J. Solids Structures 23 (1987), 1505-1520. MR 89a:73032
8. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer, New York, 1981. MR 83j:35084
9. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235-277. MR 96c:65134
10. M. Katsoulakis and A. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22 (1997), 195-233. MR 97m:35168
11. M. Katsoulakis, G. Kossioris G. and Ch. Makridakis, Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Comm. Partial Differential Equations 24 (1999), 395-424. CMP 99:11
12. C. Lattanzio and D. Serre, Convergence of a relaxation scheme for NxN hyperbolic systems of conservation laws, (preprint).
13. Y.G. Lu and C. Klingenberg, The Cauchy problem for hyperbolic conservation laws with three equations, J. Math. Anal. Appl. 202 (1996) 206-216. MR 97h:35147
14. -1,q est compacte pour tout q < 2, J. Math. Pures Appl. 60 (1981), 309-322. MR 83b:46045
15. R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), 795-823. MR 97c:35131
16. B. Perthame, An introduction to kinetic schemes for gas dynamics. In An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, D. Kroener, M. Ohlberger and C. Rohde, eds, Lecture Notes in Comp. Science and Engin., Vol. 5, Springer, Berlin, 1998, pp. 1-27.
17. D. Serre, Relaxation semi-linéaire et cinétique des systèmes de lois de conservation, Ann. Inst. H. Poincaré, Anal. Non Linéaire (to appear).
18. D. Serre and J. Shearer, Convergence with physical viscosity for nonlinear elasticity, (1993) (preprint).
19. p for the quasi-linear wave equation, Comm. Partial Differential Equations 19 (1994), 1829-1877. MR 95m:35120
20. L. Tartar, Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics - Heriot-Watt symposium - vol IV, R.J. Knops, ed, Pitman Research Notes in Mathematics, New York (1979), pp. 136-192. MR 81m:35014
21. A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of conservation laws including a relaxation term, SIAM J. Math. Anal. 28 (1997), 136-161. MR 98a:35087
22. A. Tzavaras, Viscosity and relaxation approximations for hyperbolic systems of conservation laws. In An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, D. Kroener, M. Ohlberger and C. Rohde, eds, Lecture Notes in Comp. Science and Engin., Vol. 5, Springer, Berlin, 1998, pp. 73-122.
23. A. Tzavaras, Materials with internal variables and relaxation to conservation laws, Arch. Rational Mech. Anal. 146 (1999), 129-155. CMP 2000:03