Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T18:06:41.117Z Has data issue: false hasContentIssue false
  • English
  • Français

A Jacobian-Free Newton Krylov Implicit-Explicit Time Integration Method for Incompressible Flow Problems

Published online by Cambridge University Press:  03 June 2015

Samet Y. Kadioglu and
Dana A. Knoll
Samet Y. Kadioglu*
Affiliation:
Department of Mathematical Engineering, Yildiz Technical University, 34210 Davutpasa-Esenler, Istanbul, TurkeyFuels Modeling and Simulation Department, Idaho National Laboratory, P.O. Box 1625, MS 3840, Idaho Falls, ID 83415, USA
Dana A. Knoll*
Affiliation:
Theoretical Division, Los Alamos National Laboratory, P.O. Box 1663, MS B-216, Los Alamos, NM, 87545, USA
*
Corresponding author.Email:samet.kadioglu@inl.gov
Email:nol@lanl.gov
Get access

Abstract

We have introduced a fully second order IMplicit/EXplicit (IMEX) time in-tegration technique for solving the compressible Euler equations plus nonlinear heat conduction problems (also known as the radiation hydrodynamics problems) in Kadioglu et al., J. Comp. Physics [22,24]. In this paper, we study the implications when this method is applied to the incompressible Navier-Stokes (N-S) equations. The IMEX method is applied to the incompressible flow equations in the following manner. The hyperbolic terms of the flow equations are solved explicitly exploiting the well understood explicit schemes. On the other hand, an implicit strategy is employed for the non-hyperbolic terms. The explicit part is embedded in the implicit step in such a way that it is solved as part of the non-linear function evaluation within the framework of the Jacobian-Free Newton Krylov (JFNK) method [8,29,31]. This is done to obtain a self-consistent implementation of the IMEX method that eliminates the potential order reduction in time accuracy due to the specific operator separation. We employ a simple yet quite effective fractional step projection methodology (similar to those in [11,19,21,30]) as our preconditioner inside the JFNK solver. We present results from several test calculations. For each test, we show second order time convergence. Finally, we present a study for the algorithm performance of the JFNK solver with the new projection method based preconditioner.

Keywords

01-0835Q3576DxxIncompressible flowNavier-Stokes equationsIMEX methodJFNK methodpreconditioner

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 5 , May 2013, pp. 1408 - 1431
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.)

Article purchase

Temporarily unavailable

References

[1]Armfield, S. W.Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids. Computers and Fluids, 20:1–17,1991.Google Scholar
[2]Armfield, S. W. and Street, R.Comparison of staggered and non-staggered grid Navier-Stokes solutions for the 8:1 cavity natural convection flow. ANZIAM, 46:918–934, 2004.CrossRefGoogle Scholar
[3]Ascher, U. M., Ruuth, S. J., and Spiteri, R. J.Implicit-explicit runge-kutta methods for time dependent partial differential equations. Applied Numerical Mathematics, 25:151–167, 1997.Google Scholar
[4]Ascher, U. M., Ruuth, S. J., and Wetton, B.Implicit-explicit methods for time dependent pde's. SIAM J. Numer. Anal., 32:797–823, 1995.CrossRefGoogle Scholar
[5]Batchelor, G.K.An Introduction to Fluid Dynamics. Cambridge University Press, 1967.Google Scholar
[6]Bates, J. W., Knoll, D. A., Rider, W. J., Lowrie, R. B., and Mousseau, V. A.On consistent time- integration methods for radiation hydrodynamics in the equilibrium diffusion limit: Low energy-density regime. J. Comput. Phys., 167:99–130, 2001.Google Scholar
[7]Bell, J.B., Colella, P., and Glaz, H.M.A second order projection method for the incompressible navier-stokes equations. J. Comput. Phys., 85:257–283, 1989.CrossRefGoogle Scholar
[8]Brown, P.N. and Saad, Y.Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput., 11:450–481, 1990.CrossRefGoogle Scholar
[9]Bruneau, C.H. and Saad, M.The 2-D lid-driven cavity problem revisited. Computers and Fluids, 35:326–348,2006.Google Scholar
[10]Chorin, A. and Marsden, J.E.A Mathematical Introduction to Fluid Mechanics. Springer-Verlag, 1979.Google Scholar
[11]Chorin, A. J.Numerical solution of the Navier-Stokes equations. Math. Comp., 22:742–762, 1968.Google Scholar
[12]Drikakis, D.Study of bifurcation flow phenomena in incompressible sudden expansion flows. Physics of Fluids, 9:76–87, 1997.Google Scholar
[13]Drikakis, D. and Rider, W.High-Resolution Methods for Incompressible and Low-Speed Flows. Springer., 2005.Google Scholar
[14]Dembo, R.et al. Inexact newton methods. SIAM J. Num. Anal., 19:400–408, 1982.Google Scholar
[15]Fletcher, C.A.J.Computational Techniques for Fluid Dynamics, Volume 2. Springer.Google Scholar
[16]Gottlieb, S. and Shu, C. W.Total variation diminishing runge-kutta schemes. Mathematics of Computation, 221:73–85, 1998.Google Scholar
[17]Gottlieb, S., Shu, C. W., and Tadmor, E.Strong stability-preserving high-order time discretiza-tion methods. Siam Review, 43-1:89–112, 2001.Google Scholar
[18]Griffith, B.E.An accurate and efficient method for the incompressible navier-stokes equations using the projection method as a preconditioner. J. Comput. Phys., 228:7565–7595, 2009.Google Scholar
[19]Guermond, J. L., Minev, P., and Jie Shen, . An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg., 195:6011–6045, 2006.Google Scholar
[20]Gustafson, B., Kreiss, H. O., and Oliger, J.Time dependent problems and difference methods. Wiley, New York, 1995.Google Scholar
[21]Harlow, F.H. and Welch, J.E.Numerical calculation of time dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids, 8:2182, 1965.Google Scholar
[22]Kadioglu, S. Y. and Knoll, D. A.A fully second order implicit/explicit time integration tech-nique for hydrodynamics plus nonlinear heat conduction problems. J. Comput. Phys., 229-9:3237–3249, 2010.Google Scholar
[23]Kadioglu, S. Y. and Knoll, D. A.Hydrodynamics (Advanced Topics). INTECH, 2011.Google Scholar
[24]Kadioglu, S. Y., Knoll, D. A., Lowrie, Robert B., and Rauenzahn, Rick M.A second order self-consistent IMEX method for radiation hydrodynamics. J. Comput. Phys., 229-22:8313–8332,2010.Google Scholar
[25]Kadioglu, S. Y., Knoll, D. A., and Oliveria, C.Multi-physics analysis of spherical fast burst reactors. Nuclear Science and Engineering, 163:1–12, 2009.Google Scholar
[26]Kadioglu, S. Y., Knoll, D. A., and Mark Sussman, An IMEX method for multi-phase flows. In progress, 2011.Google Scholar
[27]Kadioglu, S.Y., Knoll, D.A., Sussman, M., and Martineau, R.A second order JFNK-based IMEX method for single and multi-phase flows. Computational Fluid Dynamics, Springer-Verlag, DOI 10.1007/978-3-642-17884-9-69, 2010.Google Scholar
[28]Kadioglu, S.Y., Sussman, M., Osher, S., Wright, J.P., and Kang, M.A Second Order Primitive Preconditioner For Solving All Speed Multi-Phase Flows. J. Comput. Phys., 209-2:477–503, 2005.Google Scholar
[29]Kelley, C. T.Solving Nonlinear Equations with Newton's Method. Siam, 2003.Google Scholar
[30]Kim, J. and Moin, P.Application of a fractional-step method to incompressible navier-stokes equations. J. Comput. Phys., 59:308–323, 1985.CrossRefGoogle Scholar
[31]Knoll, D. A. and Keyes, D. E.Jacobian-free Newton Krylov methods: a survey of approaches and applications. J. Comput. Phys., 193:357–397, 2004.Google Scholar
[32]Leonard, B. P.The ultimate conservative difference scheme applied to unsteady one dimen-sional advection. Comp. Methods Appl. Mech. Eng, 88, 1991.Google Scholar
[33]Leveque, R. J.Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Texts in Applied Mathematics, 1998.Google Scholar
[34]Liu, X.D, Osher, S., and Chan, T.Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115:200–212, 1994.Google Scholar
[35]Lowrie, R. B., Morel, J. E., and Hittinger, J. A.The coupling of radiation and hydrodynamics. Astrophys. J., 521:432,1999.CrossRefGoogle Scholar
[36]Patankar, S.V.Numerical heat transfer and fluid flow. Hemisphere, New York, 1980.Google Scholar
[37]Reid, J. K.On the methods of conjugate gradients for the solution of large sparse systems of linear equations. Large Sparse Sets of Linear Equations, Academic Press, New York, 1971.Google Scholar
[38]Ruuth, S. J.Implicit-explicit methods for reaction-diffusion problems in pattern formation. J. Math. Biol., 34:148–176, 1995.Google Scholar
[39]Saad, Y.Iterative Methods for Sparse Linear Systems. Siam, 2003.Google Scholar
[40]Schreiber, R. and Keller, H.B.Driven cavity flows by efficient numerical techniques. J. Com-put. Phys., 49, 1983.Google Scholar
[41]Shu, C.W.High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Review, 51:82–126, 2009.Google Scholar
[42]Shu, C.W. and Osher, S.Efficient implementation of essentially non-oscillatory shock captur-ing schemes. J. Comput. Phys., 77:439, 1988.Google Scholar
[43]Shu, C.W. and Osher, S.Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys., 83:32, 1989.Google Scholar
[44]Strikwerda, J. C.Finite Difference Schemes Partial Differential Equations. Wadsworth & Brooks/Cole, Advance Books & Software, Pacific Grove, California, 1989.Google Scholar
[45]Sussman, M.A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles. J. Comput. Phys., 187:110–136, 2003.Google Scholar
[46]Sussman, M. and Puckett, E.G.A coupled level set and volume-of-fluid method for computing 3D and Axisymmetric incompressible two-phase flows. J. Comput. Phys., 162:301–337, 2000.Google Scholar
[47]Thomas, J.W.Numerical Partial Differential Equations I (Finite Difference Methods). Springer-Verlag New York, Texts in Applied Mathematics, 1998.Google Scholar
[48]Thomas, J.W.Numerical Partial Differential Equations II (Conservation Laws and Elliptic Equations). Springer-Verlag New York, Texts in Applied Mathematics, 1999.Google Scholar
[49]Wesseling, P.Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics, 2000.Google Scholar