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Finite difference schemes on grids with local refinement in time and space for parabolic problems I. Derivation, stability, and error analysis

Differenz-Schemata für parabolische Probleme auf Gittern mit Lokaler Verdichtung in Zeit und Raum

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Abstract

Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat).

The approximation at the grid points near the fine and coarse grid interface is based on the approach proposed by the authors in a previous paper for selfadjoint elliptic equations.

The proposed schemes are implicit of backward Euler type and are shown to be unconditionally stable. Error analysis is also presented.

Zusammenfassung

Es werden Differenzenverfahren für parabolische Anfangswertprobleme für zellenorientierte räumliche Gitter (rechteckig für zwei Raumdimensionen) mit regulärer lokaler Verfeinerung bzgl. Zeit und Raum vorgestellt. Ihre Stabilitäts- und Konvergenzeigenschaften werden studiert. Die Differenzenverfahren basieren auf der Idee der finiten Volumina durch Approximation der Bilanzgleichungen und erhalten die Masse (bzw. die Energie).

Die Approximation an den Gitterpunkten nahe der Überschneidung von feineren und gröberen Gittern verwendet eine frühere Idee der Autoren für selbstadjungierte elliptische Operatoren.

Die vorgeschlagenen Verfahren sind implizit vom Typ “Euler rückwärts” und unkonditioniert stabil. Eine Fehleranalyse ist angeschlossen.

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Authors and Affiliations

  1. Mathematics Department, University of Wyoming, University Station, P.O. Box 3036, 82071, Laramie, Wyoming, USA

    R. E. Ewing

  2. Institute of Mathematics, Bulgarian Academy of Sciences, “Acad. G. Bontchev”-street, Bl. 8, 1113, Sofia, Bulgaria

    R. D. Lazarov

  3. Center of Computer Science and Technology, bulgarian Academy of Sciences, “Acad. G. Bontchev”-street, Bl. 25a, 1113, Sofia, Bulgaria

    P. S. Vassilevski

Authors
  1. R. E. Ewing
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  2. R. D. Lazarov
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  3. P. S. Vassilevski
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Dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.

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Ewing, R.E., Lazarov, R.D. & Vassilevski, P.S. Finite difference schemes on grids with local refinement in time and space for parabolic problems I. Derivation, stability, and error analysis. Computing 45, 193–215 (1990). https://doi.org/10.1007/BF02250633

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  • DOI: https://doi.org/10.1007/BF02250633

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