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Lösung linearer Gleichungssysteme

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Numerische Strömungsmechanik

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Zusammenfassung

In diesem Kapitel werden Methoden zur Lösung der algebraischen Gleichungssysteme beschrieben, die sich aus der Diskretisierung von Transportgleichungen ergeben. Direkte Methoden werden kurz beschrieben, aber der größte Teil des Kapitels ist den iterativen Lösungstechniken gewidmet. Unvollständige LU-Zerlegung, Methoden der konjugierten Gradienten und Mehrgitterverfahren werden besonders berücksichtigt. Es werden auch Ansätze zur Lösung gekoppelter und nichtlinearer Systeme beschrieben, einschließlich der Probleme der Unterrelaxation und der Konvergenzkriterien. Verschiedene Löser können von der Buchwebseite (www.cfd-peric.de) heruntergeladen werden.

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Notes

  1. 1.

    Bini und Meini (2009) präsentieren die Geschichte der Methode der zyklischen Reduktion, ihre Erweiterungen, neue Beweise und Formeln. Sie wird als Glätter für Multigrid-Anwendungen auf hochparallelen Grafikprozessoren (GPUs) verwendet, die jetzt für die Berechnung von Fluidströmungen verwendet werden (Göddeke und Strzodka 2011).

  2. 2.

    Zum Beispiel, \(\varvec{\phi }_3=\varvec{\phi }_0 + \varvec{\rho }^0 + (I-A)\varvec{\rho }^0 + (I-A)(I-A)\varvec{\rho }^0\).

  3. 3.

    Die Diskussion in diesem Buch umfasst lineare Systeme. Kap. 14 von Shewchuk (1994) beschreibt die nichtlineare Methode der konjugierten Gradienten und ihre Vorkonditionierung.

  4. 4.

    Für eine einfache rechteckige Geometrie und Dirichlet-Randbedingungen (Brazier 1974): \(\omega =2/(1+\mathrm{sin}(\pi / N_{CV}))\).

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Author information

Authors and Affiliations

  1. Stanford University, Stanford, CA, USA

    Joel H. Ferziger

  2. Institut für Schiffstechnik, Meerestechnik und Transportsysteme (ISMT), Universität Duisburg-Essen, Erlangen, Deutschland

    Milovan Perić

  3. School of Engineering, Stanford University, Stanford, CA, USA

    Robert L. Street

Authors
  1. Joel H. Ferziger
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  2. Milovan Perić
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  3. Robert L. Street
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Correspondence to Milovan Perić .

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Ferziger, J.H., Perić, M., Street, R.L. (2020). Lösung linearer Gleichungssysteme. In: Numerische Strömungsmechanik. Springer Vieweg, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46544-8_5

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