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Non-normality effects in a discretised, nonlinear, reaction-convection-diffusion equation

Higham, D.J. and Owren, B. (1996) Non-normality effects in a discretised, nonlinear, reaction-convection-diffusion equation. Journal of Computational Physics, 124 (2). pp. 309-323. ISSN 0021-9991

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Abstract

What is the long-time effect of adding convention to a discretised reaction-diffusion equation? For linear problems, it is well known that convection may denormalise the process, and, in particular, eigenvalue-based stability predictions may be overoptimistic. This work deals with a related issue - with a nonlinear reaction term, the nonnormality can greatly influence the long-time dynamics. For a nonlinear model problem with Dirichlet boundary conditions, it is shown that the basin of attraction of the 'correct' steady state can be shrunk in a directionally biased manner. A normwise analysis provides lower bounds on the basin of attraction and a more revealing picture is provided by pseudo-eigenvalues. In extreme cases, the computed solution can converge to a spurious, bounded, steady state that exists only in finite precision arithmetic. The impact of convection on the existence and stability of spurious, periodic solutions is also quantified.

Item type: Article
ID code: 192
Keywords: discretised reaction-diffusion equation, linear problems, differential equations, mathematics, Dirichlet boundary, convection, Mathematics
Subjects: Science > Mathematics
Department: Faculty of Science > Mathematics and Statistics
Related URLs:
    Depositing user: Ms Sarah Scott
    Date Deposited: 01 Mar 2006
    Last modified: 12 Mar 2012 10:35
    URI: http://strathprints.strath.ac.uk/id/eprint/192

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