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Constant wavefront iteration methods for nine- and 15-point difference matrices

Iterative Methoden konstanten Wellenfront für neun- und fünfzehnpunktige Differenzmatrizen

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Abstract

Classical wavefront preconditioned iteration methods for difference matrices on a rectangular or on a rectangular parallelepipedal domain use wavefronts based on diagonal (line or plane, respectively) orderings of the meshpoint. Since such wavefronts do not have constant widths, they cannot be implemented efficiently on parallel computers.

We discuss various methods to get wavefronts with constant width for difference matrices for second order elliptic problems. In particular, we discuss their applications for the nine-point (2D) and 15-point (3D) difference approximations for the Laplacian, which are fourth order accurate for proper choices of the coefficients. It turns out that we can easily get preconditioning methods with wavefronts in the form of vertical or horizontal lines both in 2D and 3D, which have condition numberO(h −1), but for general three space dimensional problems no simple ordering leading to constant plane wavefronts seems to exist in general, for which the corresponding preconditioner has such a small condition number. A crucial property we make use of in the methods is the spectral equivalence between the nine-point and the standard five-point difference matrices and between the 15-point and the standard seven-point difference matrices in two and three space dimensions, respectively.

The methods use only nearest neighbor connections and can therefore be implemented efficiently not only on shared memory computers but also on distributed memory computer architectures, such as mesh-connected computer architectures.

Zusammenfassung

Klassische Wellenfront-vorkonditionierte iterative Methoden für Differenzmatrizen verwenden Wellenfronten, die auf diagonalen (Linien- oder Flächen-) Ordnungen der Gitterpunkte basieren. Da solche Wellenfronten keine konstante Breite haben, ist es nicht möglich, sie effizient auf Parallelrechner-Architekturen auszuführen.

Wir diskutieren verschiedene Methoden, wie man Wellenfronten mit konstanter Breite für elliptische Probleme zweiter Ordnung erhalten kann. Insbesondere diskutieren wir die Anwendung dieser Methoden für neun- (2D) und fünfzehnpunktige (3D) Differenzapproximationen des Laplaceoperators, die für geeignete Wahl der Koeffizienten von vierter Ordnung sind. Wir erhalten vorkonditionierte Methoden mit Wellenfronten als vertikale oder horizontale Linien sowohl in 2D als auch in 3D, die die KonditionszahlO(h −1) haben.

Die Methoden benutzen nur Verbindungen zu benachbarten Knoten. Infolgedessen können sie nicht nur auf Rechner-Architekturen mit geteiltem Speicher, sondern auch auf verteilten Systemen, Z.B. vernetzten Parallelrechner-Architekturen, effizient ausgeführt werden.

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Axelsson, O., Lindskog, G. Constant wavefront iteration methods for nine- and 15-point difference matrices. Computing 46, 233–252 (1991). https://doi.org/10.1007/BF02238300

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Key words

  • Preconditioned iterative methods
  • generalized SSOR methods
  • wavefront methods
  • 15-point difference methods
  • mesh-connected computer architectures