Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Convergence rate of approximate solutions to weakly coupled nonlinear systems

Author: Haim Nessyahu
Journal: Math. Comp. 65 (1996), 575-586
MSC (1991): Primary 35L65; Secondary 65M10, 65M15
MathSciNet review: 1333321
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled $2 \times 2$ systems appear, for example, in the context of resonant waves in gas dynamics equations.

This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or $\mathrm{Lip}^+$-stability) together with $W^{-1,1}$-consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence rate estimates, both global (quantified in terms of the $W^{s,p}$-norms) and local.

We focus our attention on the $\mathrm{Lip}^+$-stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are $\mathrm{Lip}^+$-stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of $\mathrm{Lip}^+$-stability.

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

  • 1. J.K. Hunter, "Interacting weakly nonlinear hyperbolic and dispersive waves", in Microlocal Analysis and Nonlinear Waves, M. Beals, R.B. Melrose and J. Rauch, Editors, Springer-Verlag, New York (1991). MR 92e:35103
  • 2. P.D. Lax, "Hyperbolic Systems of conservation laws and the mathematical theory of shock waves", in Regional Conf. Series Lectures in Applied Math. Vol. 11 (SIAM, Philadelphia, 1972). MR 50:2709
  • 3. A. Majda, R.R. Rosales and M. Schonbek, "A canonical system of integro-differential equations arising in resonant nonlinear acoustics", Stud. Appl. Math. Vol. 79 (1988), pp. 205-262. MR 90g:76077
  • 4. H. Nessyahu and E. Tadmor, "The convergence rate of approximate solutions for nonlinear conservation laws", Siam J. on Numer. Anal., Vol. 29 (1992), pp. 1505-1519. MR 93j:65139
  • 5. H. Nessyahu, E. Tadmor and T. Tassa, "The convergence rate of Godunov type schemes", Siam J. on Numer. Anal., Vol. 31 (1994), pp. 1-16. MR 94m:65140
  • 6. H. Nessyahu and T. Tassa, "Convergence rate of approximate solutions to conservation laws with initial rarefactions", Siam J. on Numer. Anal., Vol. 31 (1994), pp. 628-654. CMP 94:12
  • 7. O. A. Oleinik, "Discontinuous solutions of nonlinear differential equations ", Amer. Math. Soc. Transl. (2), Vol. 26 (1963), pp. 95-172. MR 20:1055;MR 27:1721
  • 8. Protter and Weinberger, "Maximum principles in Differential Equations", Prentice-Hall, Englewood Cliffs, NJ, (1967). MR 36:2935
  • 9. S. Schochet and E. Tadmor, "The regularized Chapman-Enskog expansion for scalar conservation laws", Arch. Rational Mech. Anal., Vol. 119 (1992), pp. 95-107. MR 93f:35191
  • 10. J. Smoller, "Shock Waves and Reaction-Diffusion Equations", Springer-Verlag, New York (1983). MR 84d:35002
  • 11. E. Tadmor, "Local error estimates for discontinuous solutions of nonlinear hyperbolic equations", Siam J. on Numer. Anal., Vol. 28 (1991), pp. 891-906. MR 92d:35190
  • 12. E. Tadmor, "Total variation and error estimates for spectral viscosity approximations", Math. of Comp., Vol. 60 (1993), pp. 245-256. MR 93d:35098

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 35L65, 65M10, 65M15

Retrieve articles in all journals with MSC (1991): 35L65, 65M10, 65M15

Additional Information

Received by editor(s): May 24, 1993
Additional Notes: This research was supported in part by the Basic Research Foundation, Israel Academy of Sciences and Humanities
$^{1}$ Passed away in the dawn of April the 26th, 1994, at age 29, in The Himalayas, Nepal
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society