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Mathematics of Computation
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
Published electronically: January 3, 2007
MathSciNet review: 2291834
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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.


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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: http://dx.doi.org/10.1090/S0025-5718-07-01919-9
PII: S 0025-5718(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.