Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Convergent difference schemes for the Hunter-Saxton equation


Authors: H. Holden, K. H. Karlsen and N. H. Risebro
Journal: Math. Comp. 76 (2007), 699-744
MSC (2000): Primary 35D05, 65M12; Secondary 65M06
DOI: https://doi.org/10.1090/S0025-5718-07-01919-9
Published electronically: January 3, 2007
MathSciNet review: 2291834
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We propose and analyze several finite difference schemes for the Hunter-Saxton equation

$\displaystyle u_t+u u_x= \frac12 \int_0^x (u_x)^2\,dx , \qquad x>0,\, t>0.$ (HS)

This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of $ u$, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.


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

  • 1. A. Bressan and A. Constantin.
    Global solutions of the Hunter-Saxton equation.
    SIAM J. Math. Anal. 37 (2005), 996-1026. MR 2191785 (2006j:35203)
  • 2. A. Bressan, P. Zhang, and Y. Zheng.
    Asymptotic variational wave equations.
    Arch. Rat. Mech. Anal. 183 (2007), 163-185.
  • 3. R. J. DiPerna and P.-L. Lions.
    Ordinary differential equations, transport theory and Sobolev spaces.
    Invent. Math. 98 (1989) 511-547. MR 1022305 (90j:34004)
  • 4. E. Feireisl.
    Dynamics of Viscous Compressible Fluids.
    Oxford University Press, Oxford, 2004. MR 2040667 (2005i:76092)
  • 5. H. Holden and X. Raynaud.
    A convergent numerical scheme for the Camassa-Holm equation based on multipeakons. Discrete Contin. Dyn. Syst. 14 (2006), 505-523. MR 2171724 (2006d:35226)
  • 6. J. K. Hunter and R. A. Saxton.
    Dynamics of director fields.
    SIAM J. Appl. Math. 51 (1991) 1498-1521. MR 1135995 (93a:76005)
  • 7. J. K. Hunter and Y. Zheng.
    On a completely integrable nonlinear hyperbolic variational equation.
    Physica D 79 (1994) 361-386. MR 1306466 (96e:35166)
  • 8. J. K. Hunter and Y. Zheng.
    On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions.
    Arch. Rat. Mech. Anal. 129 (1995) 305-353. MR 1361013 (96m:35215)
  • 9. J. K. Hunter and Y. Zheng.
    On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits.
    Arch. Rat. Mech. Anal. 129 (1995) 355-383. MR 1361014 (96m:35216)
  • 10. G. S. Jiang, D. Levy, C. T. Lin, S. Osher, and E. Tadmor.
    High-resolution nonoscillatory central schemes with non-staggered grids for hyperbolic conservation laws.
    SIAM J. Numer. Anal. 35 (1998) 2147-2168. MR 1655841 (99j:65145)
  • 11. J. L. Joly, G. Metivier, and J. Rauch.
    Focusing at a point and absorption of nonlinear oscillations.
    Trans. Amer. Math. Soc. 347 (1995) 3921-3969. MR 1297533 (95m:35024)
  • 12. P.-L. Lions.
    Mathematical Topics in Fluid Mechanics. Vol. 1.
    The Clarendon Press, Oxford University Press, New York, 1996.
    Incompressible models, Oxford Science Publications. MR 1422251 (98b:76001)
  • 13. P.-L. Lions.
    Mathematical Topics in Fluid Mechanics. Vol. 2.
    The Clarendon Press, Oxford University Press, New York, 1998.
    Compressible models, Oxford Science Publications. MR 1637634 (99m:76001)
  • 14. R. A. Saxton.
    Dynamic instability of the liquid crystal director.
    In Current Progress in Hyperbolic Systems: Riemann Problems and Computations (W. B. Lindquist, ed.), Contemporary Mathematics, vol. 100, American Mathematical Society, Providence, 1989, pp. 325-330. MR 1033527 (90k:35246)
  • 15. E. G. Virga.
    Variational Theories for Liquid Crystals.
    Chapman & Hall, London, 1994. MR 1369095 (97m:73001)
  • 16. P. Zhang and Y. Zheng.
    On oscillations of an asymptotic equation of a nonlinear variational wave equation.
    Asymptot. Anal. 18 (1998) 307-327. MR 1668954 (99j:35146)
  • 17. P. Zhang and Y. Zheng.
    On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation.
    Acta Math. Sinica (Engl. Ed.) 15 (1999) 115-130. MR 1701136 (2000f:35090)
  • 18. P. Zhang and Y. Zheng.
    Existence and uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data.
    Arch. Rat. Mech. Anal. 155 (2000) 49-83. MR 1799274 (2001j:35184)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 35D05, 65M12, 65M06

Retrieve articles in all journals with MSC (2000): 35D05, 65M12, 65M06


Additional Information

H. Holden
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO–7491 Trondheim, Norway, and Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway
Email: holden@math.ntnu.no

K. H. Karlsen
Affiliation: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway
Email: kennethk@math.uio.no

N. H. Risebro
Affiliation: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway
Email: nilshr@math.uio.no

DOI: https://doi.org/10.1090/S0025-5718-07-01919-9
Keywords: Hunter--Saxton equation, finite difference schemes, weak solutions, convergence, liquid crystals
Received by editor(s): November 30, 2005
Received by editor(s) in revised form: January 3, 2006
Published electronically: January 3, 2007
Additional Notes: This work was partially supported by the BeMatA program of the Research Council of Norway and the European network HYKE, contract HPRN-CT-2002-00282.
The research of the second author was supported by an Outstanding Young Investigators Award from the Research Council of Norway.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society