Convergence of Vanishing Viscosity Approximations of 2 x 2 Triangular Systems of Multi-Dimensional Conservation Laws

G. M. Coclite; K. H. Karlsen; S. Mishra; N. H. Risebro

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 1, page 275-284
  • ISSN: 0392-4041

Abstract

top
We consider a multidimensional triangular system of conservation laws. These equations arise in models of three phase flows in porous media and include multi dimensional conservation laws with discontinuous coefficients as special cases. We study approximate solutions of these equations constructed by the vanishing viscosity method and show that the approximate solutions converge to a weak solution of the multi-dimensional triangular system.

How to cite

top

Coclite, G. M., et al. "Convergence of Vanishing Viscosity Approximations of 2 x 2 Triangular Systems of Multi-Dimensional Conservation Laws." Bollettino dell'Unione Matematica Italiana 2.1 (2009): 275-284. <http://eudml.org/doc/290564>.

@article{Coclite2009,
abstract = {We consider a multidimensional triangular system of conservation laws. These equations arise in models of three phase flows in porous media and include multi dimensional conservation laws with discontinuous coefficients as special cases. We study approximate solutions of these equations constructed by the vanishing viscosity method and show that the approximate solutions converge to a weak solution of the multi-dimensional triangular system.},
author = {Coclite, G. M., Karlsen, K. H., Mishra, S., Risebro, N. H.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {275-284},
publisher = {Unione Matematica Italiana},
title = {Convergence of Vanishing Viscosity Approximations of 2 x 2 Triangular Systems of Multi-Dimensional Conservation Laws},
url = {http://eudml.org/doc/290564},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Coclite, G. M.
AU - Karlsen, K. H.
AU - Mishra, S.
AU - Risebro, N. H.
TI - Convergence of Vanishing Viscosity Approximations of 2 x 2 Triangular Systems of Multi-Dimensional Conservation Laws
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/2//
PB - Unione Matematica Italiana
VL - 2
IS - 1
SP - 275
EP - 284
AB - We consider a multidimensional triangular system of conservation laws. These equations arise in models of three phase flows in porous media and include multi dimensional conservation laws with discontinuous coefficients as special cases. We study approximate solutions of these equations constructed by the vanishing viscosity method and show that the approximate solutions converge to a weak solution of the multi-dimensional triangular system.
LA - eng
UR - http://eudml.org/doc/290564
ER -

References

top
  1. MISHRA, ADIMURTHI-SIDDHARTHA - VEERAPPA GOWDA, G. D., Optimal entropy solutions for conservation laws with discontinuous flux. Journal of Hyp. Diff. Eqns., 2 (4) (2005), 783-837. Zbl1093.35045MR2195983DOI10.1142/S0219891605000622
  2. BENZONI-GAVAGE, S. - SERRE, D., Multidimensional hyperbolic partial differential equations. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2007. First-order systems and applications. MR2284507
  3. BÜRGER, R. - KARLSEN, K. H. - RISEBRO, N. H. - TOWERS, J. D., Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units. Numer. Math., 97 (1) (2004), 25-65. Zbl1053.76047MR2045458DOI10.1007/s00211-003-0503-8
  4. CHEN, Z., Computational methods for multiphase flows in porous media. Computational science and Engineering, SIAM, 2006. Zbl1092.76001MR2217767DOI10.1137/1.9780898718942
  5. COCLITE, G. M. - MISHRA, S. - RISEBRO, N. H., Convergence of Engquist-Osher scheme for a multidimensional triangular system of conservation laws, submitted. Zbl1369.65108
  6. DIEHL, S., A conservation law with point source and discontinuous flux function modeling continuous sedimentation. SIAM J. Appl. Math., 56 (2) (1995), 1980-2007. MR1381652DOI10.1137/S0036139994242425
  7. GIMSE, T. - RISEBRO, N. H., Solution of Cauchy problem for a conservation law with discontinuous flux function. SIAM J. Math. Anal, 23 (3) (1992), 635-648. Zbl0776.35034MR1158825DOI10.1137/0523032
  8. HÖRMANDER, L., Lectures on non-linear hyperbolic differential equations. Mathematics and Applications, Vol 26, Springer, 1997. MR1466700
  9. KARLSEN, K. H. - MISHRA, S. - RISEBRO, N. H., Convergence of finite volume schemes for triangular systems of conservation laws. Submitted. Zbl1190.65133MR2471610DOI10.1007/s00211-008-0199-x
  10. KARLSEN, K. H. - RASCLE, M. - TADMOR, E., On the existence and compactness of a two-dimensional resonant system of conservation laws. Commun. Math. Sci., 5 (2), 2007. Zbl1165.35412MR2334842
  11. KARLSEN, K. H. - RISEBRO, N. H. - TOWERS, J. D., L 1 stability for entropy solution of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk., 3 (2003), 49 pages. Zbl1036.35104MR2024741
  12. KARLSEN, K. H. - TOWERS, J. D., Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux. Chinese Ann. Math. Ser. B, 25 (3) (2004), 287-318. Zbl1112.65085MR2086124DOI10.1142/S0252959904000299
  13. MURAT, F., L'injection du cône positif de H - 1 dans W - 1 , q est compacte pour tout q < 2 . J. Math. Pures Appl. (9), 60 (3), (1981), 309-322. Zbl0471.46020MR633007
  14. PANOV, E. YU., Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux. Submitted. Zbl1200.35052MR2592291DOI10.1007/s00205-009-0217-x
  15. TOWERS, J. D., Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal., 38 (2) (2000), 681-698. Zbl0972.65060MR1770068DOI10.1137/S0036142999363668
  16. TOWERS, J. D., A difference scheme for conservation laws with a discontinuous flux, the nonconvex case. SIAM J. Numer. Anal., 39 (4) (2001), 1197-1218. Zbl1055.65104MR1870839DOI10.1137/S0036142900374974

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.