Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Convergence of relaxation schemes to the equations of elastodynamics


Authors: Laurent Gosse and Athanasios E. Tzavaras
Journal: Math. Comp. 70 (2001), 555-577
MSC (2000): Primary 35L65, 65M12
Published electronically: March 24, 2000
MathSciNet review: 1813140
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We study the effect of approximation matrices to semi-discrete relaxation schemes for the equations of one-dimensional elastodynamics. We consider a semi-discrete relaxation scheme and establish convergence using the $L^p$ theory of compensated compactness. Then we study the convergence of an associated relaxation-diffusion system, inspired by the scheme. Numerical comparisons of fully-discrete schemes are carried out.


References [Enhancements On Off] (What's this?)

  • 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. Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
  • 4. Gui Qiang Chen, C. David Levermore, and Tai-Ping Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), no. 6, 787–830. MR 1280989, 10.1002/cpa.3160470602
  • 5. Frédéric Coquel and Benoît Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal. 35 (1998), no. 6, 2223–2249 (electronic). MR 1655844, 10.1137/S0036142997318528
  • 6. R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, 10.1007/BF00251724
  • 7. C. Făciu and M. Mihăilescu-Suliciu, The energy in one-dimensional rate-type semilinear viscoelasticity, Internat. J. Solids Structures 23 (1987), no. 11, 1505–1520. MR 918435, 10.1016/0020-7683(87)90066-7
  • 8. Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
  • 9. Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276. MR 1322811, 10.1002/cpa.3160480303
  • 10. Markos A. Katsoulakis and Athanasios E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22 (1997), no. 1-2, 195–233. MR 1434144, 10.1080/03605309708821261
  • 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 $N\times N$ hyperbolic systems of conservation laws, (preprint).
  • 13. Yun-guang Lu and Christian Klingenberg, The Cauchy problem for hyperbolic conservation laws with three equations, J. Math. Anal. Appl. 202 (1996), no. 1, 206–216. MR 1402598, 10.1006/jmaa.1996.0313
  • 14. François Murat, L’injection du cône positif de 𝐻⁻¹ dans 𝑊^{-1,𝑞} est compacte pour tout 𝑞<2, J. Math. Pures Appl. (9) 60 (1981), no. 3, 309–322 (French, with English summary). MR 633007
  • 15. Roberto Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), no. 8, 795–823. MR 1391756, 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3
  • 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 Linèaire (to appear).
  • 18. D. Serre and J. Shearer, Convergence with physical viscosity for nonlinear elasticity, (1993) (preprint).
  • 19. James W. Shearer, Global existence and compactness in 𝐿^{𝑝} for the quasi-linear wave equation, Comm. Partial Differential Equations 19 (1994), no. 11-12, 1829–1877. MR 1301175
  • 20. L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
  • 21. Aslak Tveito and Ragnar Winther, On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28 (1997), no. 1, 136–161. MR 1427731, 10.1137/S0036141094263755
  • 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

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 35L65, 65M12

Retrieve articles in all journals with MSC (2000): 35L65, 65M12


Additional Information

Laurent Gosse
Affiliation: Foundation for Research and Technology Hellas / Institute of Applied and Computational Mathematics, P.O. Box 1527, 71110 Heraklion, Crete, Greece
Email: laurent@palamida.math.uch.gr

Athanasios E. Tzavaras
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: tzavaras@math.wisc.edu

DOI: https://doi.org/10.1090/S0025-5718-00-01256-4
Keywords: Relaxation schemes, compensated compactness
Received by editor(s): March 23, 1999
Published electronically: March 24, 2000
Additional Notes: This joint work was partially supported by TMR project HCL #ERBFMRXCT960033. The second author acknowledges support of the National Science Foundation and the Office for Naval Research
Article copyright: © Copyright 2000 American Mathematical Society