Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T09:12:04.539Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Published online by Cambridge University Press:  03 June 2015

E. Ferrer ,
D. Moxey ,
R. H. J. Willden  and
S. J. Sherwin
E. Ferrer*
Affiliation:
ETSIA – School of Aeronautics, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3, Madrid, E-28040, Spain
D. Moxey*
Affiliation:
Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK
R. H. J. Willden*
Affiliation:
Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK
S. J. Sherwin*
Affiliation:
Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK
*
Corresponding author.Email:esteban.ferrer@upm.es
Email:d.moxey@imperial.ac.uk
Email:richard.willden@eng.ox.ac.uk
Email:s.sherwin@imperial.ac.uk
Get access

Abstract

This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used.

Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.

Keywords

76D0535Q3065N30Incompressible Navier-Stokes equationsprojection methodsvelocity-correctioncontinuous Galerkindiscontinuous Galerkininf-sup LBB condition
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 3 , September 2014 , pp. 817 - 840
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arnold, D.N., Brezzi, F., Cockburn, B., and Marini, L.D.Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal of Numerical Analysis, 39(5):1749–1779, 2001.Google Scholar
[2]Auteri, F., Guermond, J.L., and Parolini, N.Role of the LBB condition in weak spectral projection methods: 405. spournal of Computational Physics, 174(1):405–420,2001.CrossRefGoogle Scholar
[3]Babuška, I.The finite element method with Lagrangian multipliers. Numerische Mathematik, 20(3):179–192,1973.Google Scholar
[4]Badia, S. and Codina, R.Algebraic pressure segregation methods for the incompressible Navier-Stokes equations. Archives of Computational Methods in Engineering, 15:1–52, 2007.Google Scholar
[5]Brezzi, F.An the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. ESAIM: Mathematical Modelling and Numerical Analysis, 8:129–151,1974.Google Scholar
[6]Brezzi, F. and Bathe, K.J.A discourse on the stability conditions for mixed finite element formulations. Computational Methods in Applied Mechanics and Engineering, 82:27–57, 1990.Google Scholar
[7]Brezzi, F., Cockburn, B., Marini, L.D., and Süli, E.Stabilization mechanisms in discontinuous Galerkin finite element methods. Computer Methods in Applied Mechanics and Engineering, 195(25–28):3293–3310, 2006.CrossRefGoogle Scholar
[8]Chorin, A.J.Numerical solution of the Navier-Stokes equations. Mathematics of Computation, 22:742–762,1968.CrossRefGoogle Scholar
[9]Cockburn, B., Kanschat, G., and Schötzau, D.An equal-order DG method for the incompressible Navier-Stokes equations. Journal of Scientific Computing, 40:188–210,2009.Google Scholar
[10]Codina, R.Pressure stability in fractional step finite element methods for incompressible flows. Journal of Computational Physics, 140(1):112–140,2001.Google Scholar
[11]Deville, M.O., Fischer, P.F., and Mund, E.H.High-order methods for incompressible fluid flow. Cambridge monographs on applied and computational mathematics. Cambridge University Press, 2002.Google Scholar
[12]Duffy, M.G.Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM Journal on Numerical Analysis, 19(6):1260–1262,1982.Google Scholar
[13]Elman, H.C., Silvester, D.J., and Wathen, A.J.Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, 2005.Google Scholar
[14]Epshteyn, Y. and Rivière, B.Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206:843–872, September 2007.Google Scholar
[15]Ferrer, E.A high order Discontinuous Galerkin - Fourier incompressible 3D Navier-Stokes solver with rotating sliding meshes for simulating cross-flow turbines. PhD thesis, University of Oxford, 2012.Google Scholar
[16]Ferrer, E. and R. Willden, H.J.A high order discontinuous Galerkin - Fourier incompressible 3D Navier-Stokes solver with rotating sliding meshes. Journal of Computational Physics, 231(21):7037–7056,2012.Google Scholar
[17]Ferrer, E. and Willden, R.H.J.A high order discontinuous Galerkin finite element solver for the incompressible Navier-Stokes equations. Computers & Fluids, 46(1):224–230,2011.Google Scholar
[18]Guermond, J.L., Minev, P., and Shen, J.An overview of projection methods for incompressible flows. Computer Methods in Applied Mechanics and Engineering, 195(44–47):6011–6045, 2006.CrossRefGoogle Scholar
[19]Guermond, J.L. and Quartapelle, L.An stability and convergence of projection methods based on pressure Poisson equation. International Journal of Numerical Methods in Fluids, 26:1039–1053,1998.Google Scholar
[20]Guermond, J.L. and Shen, J.Velocity-correction projection methods for incompressible flows. SIAM Journal of Numerical Analysis, 41:112–134, January 2003.Google Scholar
[21]Hartmann, R.Numerical analysis of higher order discontinuous Galerkin finite element methods. In Deconinck, H., editor, VKI LS 2008-08: CFD - ADIGMA course on very high order discretization methods, Oct. 13–17, 2008. yon Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium, 2008.Google Scholar
[22]Hu, Ning, Guo, Xian-Zhong, and Kat, I. NormanBounds, z.for eigenvalues and condition numbers in the p-version of the finite element method. Mathematics of Computation, 67(224):1423–1450,1998.CrossRefGoogle Scholar
[23]Johnson, C.Numerical solutions of partial differential equations by the finite element method. Cambridge University Press, 1987.Google Scholar
[24]Karniadakis, G.E., Israeli, M., and Orszag, S.A.High-order splitting methods for incompressible Navier-Stokes equations. Journal of Computational Physics, 97:414–443,1991.Google Scholar
[25]Karniadakis, G.E. and Sherwin, S.J.Spectral h/p element methods for computational fluid dynamics. Oxford Science Publications, 2005.Google Scholar
[26]Ladyzhenskaya, O.A.The mathematical theory of viscous incompressible flow. Gordon and Breach, New York, 2nd edition, 1969.Google Scholar
[27]Maday, Y., Meiron, D., Patera, A., and Ronquist, E.Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations. SIAM Journal on Scientific Computing, 14(2):310–337,1993.Google Scholar
[28]Maday, Y., Patera, A., and Rønquist, E.M.An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow. Journal of Scientific Computing, 5(4):263–292,1990.CrossRefGoogle Scholar
[29]Orszag, S.A., Israeli, M., and Deville, M.Boundary conditions for incompressible flows. Journal of Scientific Computing, 1(1):75–111,1986.Google Scholar
[30]Rannacher, R.An Chorin’s projection method for the incompressible Navier-Stokes equations. volume 1530 of Lecture Notes in Mathematics, pages 167–183. sppringer Berlin Heidelberg, 1992.Google Scholar
[31]Rivière, B.Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.Google Scholar
[32]Shahbazi, K., Fischer, P.F., and Ethier, C.R.A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations. Journal of Computational Physics, 222(1):391–407, 2007.Google Scholar
[33]Shen, J.Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations. In Guo, B, editor, Proceedings of the 1994 Beijing Symposium on Nonlinear Evolution Equations and Infinite Dynamical Systems, pages 68–78. sphongShan University Press, 1997.Google Scholar
[34]Steinmoeller, D.T., Stastna, M., and Lamb, K.G.A short note on the discontinuous galerkin discretization of the pressure projection operator in incompressible flow. Journal of Computational Physics, 251(0):480–486, 2013.Google Scholar
[35]Temam, A.R.Une methodé d’approximation de la solution des équations de Navier-Stokes. Bulletin de la Société Mathematique de France, 96:115–152,1968.Google Scholar