### Abstract

Language | English |
---|---|

Pages | 199-229 |

Number of pages | 30 |

Journal | Computing |

Volume | 66 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2001 |

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### Keywords

- artificial viscosity
- boundary layer
- convection-diffusion
- singular perturbation
- uniform convergence

### Cite this

*Computing*,

*66*(2), 199-229. https://doi.org/10.1007/s006070170035

}

*Computing*, vol. 66, no. 2, pp. 199-229. https://doi.org/10.1007/s006070170035

**Fundamental systems of numerical schemes for linear convection diffusion equations and their relationship to accuracy.** / Ainsworth, Mark; Dörfler, Willy.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Fundamental systems of numerical schemes for linear convection diffusion equations and their relationship to accuracy

AU - Ainsworth, Mark

AU - Dörfler, Willy

PY - 2001/3

Y1 - 2001/3

N2 - A new approach towards the assessment and derivation of numerical methods for convection dominated problems is presented, based on the comparison of the fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci.

AB - A new approach towards the assessment and derivation of numerical methods for convection dominated problems is presented, based on the comparison of the fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci.

KW - artificial viscosity

KW - boundary layer

KW - convection-diffusion

KW - singular perturbation

KW - uniform convergence

UR - http://dx.doi.org/10.1007/s006070170035

U2 - 10.1007/s006070170035

DO - 10.1007/s006070170035

M3 - Article

VL - 66

SP - 199

EP - 229

JO - Computing

T2 - Computing

JF - Computing

SN - 0010-485X

IS - 2

ER -