## Abstract

Moffatt and Duffy [1] have shown that the solution to the Poisson equation, defined on rectangular domains, includes a local similarity term of the form: r^{2}log(r)cos(2θ). The latter means that the second (and higher) derivative of the solution with respect to r is singular at r = 0. Standard high-order numerical schemes require the existence of high-order derivatives of the solution. Thus, for the case considered by Moffatt and Duffy, the high-order finite-difference schemes loose their high-order convergence due to the nonregularity at r = 0. In this article, a simple method is outlined to regain the high-order accuracy of these schemes, without the need of any modification in the scheme's algorithm. This is a significant consideration when one wants to use a given finite-difference computer code for problems with local nonregular similarity solutions. Numerical examples using the modified scheme in conjunction with a sixth-order finite difference approximation are provided.

Original language | English |
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Pages (from-to) | 336-346 |

Number of pages | 11 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jul 2001 |

## Keywords

- High-order scheme
- Local nonregular similarity
- Poisson's equation

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics