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.

ORCID iDs

Higham, D.J. ORCID logoORCID: https://orcid.org/0000-0002-6635-3461 and Owren, B.;
  • Item type:Article
    ID code:192
    Dates:
    Date
    Event
    15 March 1996
    Published
    Keywords:discretised reaction-diffusion equation, linear problems, differential equations, mathematics, Dirichlet boundary, convection, Mathematics, Physics and Astronomy (miscellaneous), Computer Science Applications
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Ms Sarah Scott
    Date deposited:01 Mar 2006
    Last modified:20 Jan 2021 16:42
    URI:https://strathprints.strath.ac.uk/id/eprint/192
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