### Abstract

Original language | English |
---|---|

Pages (from-to) | 309-323 |

Number of pages | 14 |

Journal | Journal of Computational Physics |

Volume | 124 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Mar 1996 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Computational Physics*,

*124*(2), 309-323. https://doi.org/10.1006/jcph.1996.0062

}

*Journal of Computational Physics*, vol. 124, no. 2, pp. 309-323. https://doi.org/10.1006/jcph.1996.0062

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Non-normality effects in a discretised, nonlinear, reaction-convection-diffusion equation

AU - Higham, D.J.

AU - Owren, B.

PY - 1996/3/15

Y1 - 1996/3/15

N2 - 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.

AB - 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.

KW - discretised reaction-diffusion equation

KW - linear problems

KW - differential equations

KW - mathematics

KW - Dirichlet boundary

KW - convection

UR - http://dx.doi.org/doi:10.1006/jcph.1996.0062

U2 - 10.1006/jcph.1996.0062

DO - 10.1006/jcph.1996.0062

M3 - Article

VL - 124

SP - 309

EP - 323

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -