Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-06T04:24:28.623Z Has data issue: false hasContentIssue false
  • English
  • Français

On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD

Published online by Cambridge University Press:  03 June 2015

H. Vitoshkin  and
A. Yu. Gelfgat
H. Vitoshkin*
Affiliation:
School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Ramat Aviv 69978, Tel-Aviv, Israel
A. Yu. Gelfgat
Affiliation:
School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Ramat Aviv 69978, Tel-Aviv, Israel
*
Corresponding author.Email:gelfgat@tau.ac.il
Get access

Abstract

Factorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

Keywords

76D05CFDNewton iterationArnoldi iterationStokes operator
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 4 , October 2013 , pp. 1103 - 1119
Copyright
Copyright © Global Science Press Limited 2013

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]Tuckerman, L.S., Barkley, D., Bifurcation analysis for time-steppers, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, Doedel, K. and Tuckerman, L. (Eds.), IMA Volumes in Mathematics and Its Applications, vol. 119 (2000), Springer, New York, pp. 453–466.Google Scholar
[2]Tuckerman, L.S., Bertagnolio, F., Daube, O., Quéré, P.Le, Barkley, D., Stokes preconditioning for the inverse Arnoldi method, in Continuation Methods for Fluid Dynamics, Henry, D. and Bergeon, A. (Eds.), Notes on Numerical Fluid Dynamics, vol. 74 (2000), Vieweg, Göttingen, pp. 241–255.Google Scholar
[3]Boronńska, K., Tuckerman, L.S., Extreme multiplicity in cylindrical Rayleigh-Bénard convection. I. Time dependence and oscillations, Phys. Rev. E, 81 (2010), 036320.CrossRefGoogle Scholar
[4]Boronńska, K., Tuckerman, L. S., Extreme multiplicity in cylindrical Rayleigh-Bénard convection. II. Bifurcation diagram and symmetry classification, Phys. Rev. E, 81 (2010), 036321.Google Scholar
[5]Beaume, C., Bergeon, A., Knobloch, E., Homoclinic snaking of localized states in doubly diffusive convection, Phys. Fluids, 23 (2011), 094102.Google Scholar
[6]Lu, L., Papadakis, G., Investigation of the effect of external periodic flow pulsation on a cylinder wake using linear stability analysis, Phys. Fluids, 23 (2011), 094105.Google Scholar
[7]Garnaud, X., Lesshaft, L., Schmid, P.J., Chomaz, J.-M., A relaxation method for large eigenvalue problems, with an application to flow stability analysis, J. Comput. Phys., 231 (2012), 3912–3927.Google Scholar
[8]Barkley, D., Blackburn, H.M., Sherwin, S.J., Direct optimal growth analysis for timesteppers, Int. J.Numer. Meths. Fluids, 57 (2008), 1435–1458CrossRefGoogle Scholar
[9]Yu, A.Gelfgat, Stability of convective flows in cavities: Solution of benchmark problems by a low-order finite volume method, Int. J.Numer. Meths. Fluids, 53 (2007), 485–506.Google Scholar
[10]Feldman, Yu., Direct numerical simulation of transitions and supercritical regimes in confined three dimensional recirculating flows, PhD Thesis, Tel-Aviv University, 2011.Google Scholar
[11]Feldman, Yu., Yu, A.Gelfgat, On pressure-velocity coupled time-integration of incompressible Navier-Stokes equations using direct inversion of Stokes operator or accelerated multi-grid technique, Comput. Struct., 87 (2009), 710–720.Google Scholar
[12]Deville, M.O., Fischer, P.F., Mund, E.H., High-Order Methods for Incompressible Fluid Flow, Cambridge, 2002.Google Scholar
[13]Lynch, R.E., Rice, J.R., Thomas, D.H., Direct solution of partial differential equations by tensor product methods, Numer. Math., 6 (1964), 185–199.Google Scholar
[14]Peyret, R., Spectral Methods for Incompressible Viscous Flows, Springer, 2002.Google Scholar
[15]Tric, E., Labrosse, G., Betrouni, M., A first incursion into the 3D structure of natural convection of air in a differentially heated cavity, from accurate numerical solutions, Int. J.Heat Mass Transf., 43 (1999), 4043–4056.Google Scholar
[16]Herrada, M. A., Del Pino, C., Ortega-Casanova, J., Confined swirling jet impingement on a flat plate at moderate Reynolds numbers, Phys. Fluids, 21 (2009), 013601.CrossRefGoogle Scholar
[17]Barbosa, E., Daube, O., A finite difference method for 3D incompressible flows in cylindrical coordinates, Computers & Fluids, 34 (2005), 950–971.Google Scholar
[18]Vedy, E., Viazzo, S., Schiestel, R., A high-order finite difference method for incompressible fluid turbulence simulations, Int. J.Numer. Meths. Fluids, 42 (2003), 1155–1188.Google Scholar
[19]Bjøntegaard, T., Maday, Y., Rønquist, E.M., Fast tensor-product solvers: Partially deformed three-dimensional domains, J. Sci. Comput., 39 (2009), 28–48.Google Scholar
[20]Bacuta, C., A unified approach for Uzawa algorithms, SIAM J. Numer. Anal., 44 (2006), 2633–2649.CrossRefGoogle Scholar
[21]Patankar, S.V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.Google Scholar
[22]Ni, M.-J., Abdou, M.A., A bridge between projection methods and SIMPLE type methods for incompressible Navier-Stokes equations, Int. J.Numer. Meths. Fluids, 72 (2007), 1490–1512.Google Scholar
[23] G. de Vahl Davis, Jones, I.P., Natural convection in a square cavity: A comparison exercise, Intl. J. Numer. Meths. Fluids, 3 (1983), 227–264.Google Scholar
[24]Quéré, P. Le, Accurate solutions to the square thermally driven cavity at high Rayleigh number, Computers & Fluids, 20 (1991), 29–41.Google Scholar
[25]Christon, M.A., Gresho, P.M., Sutton, S.B., Computational predictability of time-dependent natural convection flows in enclosures (including a benchmark solution), Int. J.Numer. Meths. Fluids, 40 (2002), 953–980.Google Scholar