Stability analysis for different formulations of the nonlinear term in PN−PN−2 spectral element discretizations of the navier–stokes equations

Kleiser, Leonhard; Wilhelm, Dirk

doi:10.1006/jcph.2001.6912

Publication type:	Article in scientific journal
Type of review:	Peer review (publication)
Title:	Stability analysis for different formulations of the nonlinear term in PN−PN−2 spectral element discretizations of the navier–stokes equations
Authors:	Kleiser, Leonhard Wilhelm, Dirk
DOI:	10.1006/jcph.2001.6912
Published in:	Journal of Computational Physics
Volume(Issue):	174
Issue:	1
Page(s):	306
Pages to:	326
Issue Date:	Nov-2001
Publisher / Ed. Institution:	Elsevier
ISSN:	0021-9991 1090-2716
Language:	English
Subject (DDC):	510: Mathematics
Abstract:	We show that for the PN−PN−2 spectral element method, in which velocity and pressure are approximated by polynomials of order N and N−2, respectively, numerical instabilities can occur in the spatially discretized Navier–Stokes equations. Both a staggered and nonstaggered arrangement of the N−2 pressure points are considered. These instabilities can be masked by viscous damping at low Reynolds numbers. We demonstrate that the instabilities depend on the formulation of the nonlinear term. The numerical discretization is stable for the convective form but unstable for the divergence and the skew-symmetric form. Further numerical analysis indicates that this instability is not caused by nonlinear effects, since it occurs for linearized systems as well. An eigenvalue analysis of the fully discretized system shows that an instability is introduced by the formulation of the nonlinear term. We demonstrate that the instability is related to the divergence error of the computed solution at those velocity points at which the continuity equation is not enforced.
URI:	https://digitalcollection.zhaw.ch/handle/11475/13735
Fulltext version:	Published version
License (according to publishing contract):	Licence according to publishing contract
Departement:	School of Engineering
Appears in collections:	Publikationen School of Engineering

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Kleiser, L., & Wilhelm, D. (2001). Stability analysis for different formulations of the nonlinear term in PN−PN−2 spectral element discretizations of the navier–stokes equations. Journal of Computational Physics, 174(1), 306–326. https://doi.org/10.1006/jcph.2001.6912

Kleiser, L. and Wilhelm, D. (2001) ‘Stability analysis for different formulations of the nonlinear term in PN−PN−2 spectral element discretizations of the navier–stokes equations’, Journal of Computational Physics, 174(1), pp. 306–326. Available at: https://doi.org/10.1006/jcph.2001.6912.

L. Kleiser and D. Wilhelm, “Stability analysis for different formulations of the nonlinear term in PN−PN−2 spectral element discretizations of the navier–stokes equations,” Journal of Computational Physics, vol. 174, no. 1, pp. 306–326, Nov. 2001, doi: 10.1006/jcph.2001.6912.

KLEISER, Leonhard und Dirk WILHELM, 2001. Stability analysis for different formulations of the nonlinear term in PN−PN−2 spectral element discretizations of the navier–stokes equations. Journal of Computational Physics. November 2001. Bd. 174, Nr. 1, S. 306–326. DOI 10.1006/jcph.2001.6912

Kleiser, Leonhard, and Dirk Wilhelm. 2001. “Stability Analysis for Different Formulations of the Nonlinear Term in PN−PN−2 Spectral Element Discretizations of the Navier–Stokes Equations.” Journal of Computational Physics 174 (1): 306–26. https://doi.org/10.1006/jcph.2001.6912.

Kleiser, Leonhard, and Dirk Wilhelm. “Stability Analysis for Different Formulations of the Nonlinear Term in PN−PN−2 Spectral Element Discretizations of the Navier–Stokes Equations.” Journal of Computational Physics, vol. 174, no. 1, Nov. 2001, pp. 306–26, https://doi.org/10.1006/jcph.2001.6912.