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

D.J. Higham, B. Owren

Research output: Contribution to journalArticle

5 Citations (Scopus)

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.
LanguageEnglish
Pages309-323
Number of pages14
JournalJournal of Computational Physics
Volume124
Issue number2
DOIs
Publication statusPublished - 15 Mar 1996

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convection-diffusion equation
attraction
convection
eigenvalues
reaction-diffusion equations
Boundary conditions
boundary conditions
predictions
Convection

Keywords

  • discretised reaction-diffusion equation
  • linear problems
  • differential equations
  • mathematics
  • Dirichlet boundary
  • convection

Cite this

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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.",
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Non-normality effects in a discretised, nonlinear, reaction-convection-diffusion equation. / Higham, D.J.; Owren, B.

In: Journal of Computational Physics, Vol. 124, No. 2, 15.03.1996, p. 309-323.

Research output: Contribution to journalArticle

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