Skip Navigation


IMA Journal of Numerical Analysis Advance Access originally published online on May 26, 2008
IMA Journal of Numerical Analysis 2009 29(3):539-572; doi:10.1093/imanum/drn026
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
29/3/539    most recent
drn026v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Holden, H
Right arrow Articles by Risebro, N. H.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A convergent finite-difference method for a nonlinear variational wave equation

H Holden{dagger}

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway and Centre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway

K. H. Karlsen{ddagger} and N. H. Risebro§

Centre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway

{dagger} Corresponding author. Email: holden{at}math.ntnu.no

{ddagger} Email: kennethk{at}math.uio.no

§ Email: nilshr{at}math.uio.no

Received on 20 August 2007. Revised on 29 January 2008.


  Abstract

We establish rigorously convergence of a semidiscrete upwind scheme for the nonlinear variational wave equation utt – c(u)(c(u)ux)x = 0 with u|t=0 = u0 and ut|t=0 = v0. Introducing Riemann invariants R = ut + cux and S = ut – cux, the variational wave equation is equivalent to Rt – cRx Formula (R2 – S2) and St + cSx = –Formula (R2 – S2) with Formula = c'/(4c). An upwind scheme is defined for this system. We assume that the speed c is positive, increasing and both c and its derivative are bounded away from zero and that R|t=0, S|t=0 isin L1(R) {cap} L3(R) are nonpositive. The numerical scheme is illustrated on several examples.

Key Words: variational wave equation; convergence of finite-difference schemes; liquid crystals


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.