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IMA Journal of Numerical Analysis Advance Access published online on May 26, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn026
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© 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

KH. Karlsen{ddagger} and NH. Risebro§

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

{dagger} 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 Formula with Formula and Formula . Introducing Riemann invariants R = ut + cux and S = ut – cux, the variational wave equation is equivalent to Formula and Formula with Formula . 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 Formula are nonpositive. The numerical scheme is illustrated on several examples.

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


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