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Analysis of the convergence of iterative implicit and defect-correction algorithms for hyperbolic problems

Abstract : This paper studies the convergence of unfactored implicit schemes for the solution of the steady discrete Euler equations. In these schemes first and second order accurate discretisations are simultaneously used. The close resemblance of these schemes with iterative defect correction is shown. Linear model problems are introduced for the one-dimensional and the two-dimensional cases. These model problems are analyzed in detail both by Fourier and by matrix analyses. The convergence behaviour appears to be strongly dependent on a parameter b that determines the amount of upwinding in the discretisation of the second order scheme. In general, in the iteration, after an impulsive phase a slower pseudo-convective (or Fourier) phase can be distinguished and finally again a faster asymptotic phase. The extreme parameter values b = 0 (no upwinding) and b = 1 (full second order upwinding) both appear as special cases for which the convergence behaviour degenerates. They are not recommended for practical use. For the intermediate value of b the pseudo-convection phase is less significant. Fromm's scheme (b = 1/2) or Van Leer's third order scheme (b = 1/3) show a quite satisfactory convergence behaviour. In this paper, first the linear convection problem in one and two dimensions is studied in detail. Differences between the various cases are signalized. In the last section experiments are shown for the Euler equations, including comments on how the theory is well or partially verified depending on the problem.
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https://hal.inria.fr/inria-00075358
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 6:03:42 PM
Last modification on : Saturday, January 27, 2018 - 1:30:57 AM
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Jean-Antoine Desideri, Pieter W. Hemker. Analysis of the convergence of iterative implicit and defect-correction algorithms for hyperbolic problems. [Research Report] RR-1200, INRIA. 1990. ⟨inria-00075358⟩

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