December  2012, 5(6): 1195-1221. doi: 10.3934/dcdss.2012.5.1195

A framework for the development of implicit solvers for incompressible flow problems

1. 

School of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom, United Kingdom, United Kingdom

Received  December 2011 Revised  March 2012 Published  August 2012

This survey paper reviews some recent developments in the design of robust solution methods for the Navier--Stokes equations modelling incompressible fluid flow. There are two building blocks in our solution strategy. First, an implicit time integrator that uses a stabilized trapezoid rule with an explicit Adams--Bashforth method for error control, and second, a robust Krylov subspace solver for the spatially discretized system. Numerical experiments are presented that illustrate the effectiveness of our generic approach. It is further shown that the basic solution strategy can be readily extended to more complicated models, including unsteady flow problems with coupled physics and steady flow problems that are nondeterministic in the sense that they have uncertain input data.
Citation: David J. Silvester, Alex Bespalov, Catherine E. Powell. A framework for the development of implicit solvers for incompressible flow problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1195-1221. doi: 10.3934/dcdss.2012.5.1195
References:
[1]

Uri M. Ascher, Steven J. Ruuth and Brian T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations,, SIAM J. Numer. Anal., 32 (1995), 797. doi: 10.1137/0732037. Google Scholar

[2]

Alexei Bespalov, Catherine E. Powell and David Silvester, A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data.,, MIMS Eprint 2011.91, (2011). Google Scholar

[3]

Jonathan Boyle, Milan Mihajlović and Jennifer Scott, HSL_MI20: an efficient {AMG} preconditioner for finite element problems in 3D,, Internat. J. Numer. Methods Engrg, 82 (2010), 64. Google Scholar

[4]

H. Damanik, J. Hron, A. Ouazzi and S. Turek, A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes,, J. Comput. Phys., 228 (2009), 3869. doi: 10.1016/j.jcp.2009.02.024. Google Scholar

[5]

Philip G. Drazin, "Introduction to Hydrodynamic Stability,'', Cambridge University Press, (2002). Google Scholar

[6]

Howard Elman, Milan Mihajlović and David Silvester, Fast iterative solvers for buoyancy driven flow problems,, J. Comput. Phys., 230 (2011), 3900. doi: 10.1016/j.jcp.2011.02.014. Google Scholar

[7]

Howard C. Elman, Alison Ramage and David J. Silvester, Algorithm 866: IFISS, a MATLAB toolbox for modelling incompressible flow,, ACM Trans. Math. Softw., 33 (2007), 2. Google Scholar

[8]

Howard C. Elman, David J. Silvester and Andrew J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations,, Numer. Math., 90 (2002), 665. doi: 10.1007/s002110100300. Google Scholar

[9]

Howard C. Elman, David J. Silvester and Andrew J. Wathen, "Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics,'', Oxford University Press, (2005). Google Scholar

[10]

Howard C. Elman and Ray S. Tuminaro, Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations,, Electron. Trans. Numer. Anal., 35 (2009), 257. Google Scholar

[11]

Oliver G. Ernst, Catherine E. Powell, David J. Silvester and Elisabeth Ullmann, Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data,, SIAM J. Sci. Comput., 31 (2009), 1424. doi: 10.1137/070705817. Google Scholar

[12]

P. M. Gresho, D. K. Gartling, J. R. Torczynski, K. A. Cliffe, K. H. Winters, T. J. Garratt, A. Spence and J. W. Goodrich, Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re=800 stable?,, Internat. J. Numer. Methods Fluids, 17 (1993), 501. doi: 10.1002/fld.1650170605. Google Scholar

[13]

Philip M. Gresho, David F. Griffiths and David J. Silvester, Adaptive time-stepping for incompressible flow; Part I: Scalar advection-diffusion,, SIAM J. Sci. Comput., 30 (2008), 2018. doi: 10.1137/070688018. Google Scholar

[14]

P. M. Gresho and R. L. Sani, "Incompressible Flow and the Finite Element Method: Volume 2: Isothermal Laminar Flow,'', John Wiley, (1998). Google Scholar

[15]

David A. Kay, Philip M. Gresho, David F. Griffiths and David J. Silvester, Adaptive time-stepping for incompressible flow; Part II: Navier-Stokes equations,, SIAM J. Sci. Comput., 32 (2010), 111. doi: 10.1137/080728032. Google Scholar

[16]

David Kay, Daniel Loghin and Andrew Wathen, A preconditioner for the steady-state Navier-Stokes equations,, SIAM J. Sci. Comput., 24 (2002), 237. doi: 10.1137/S106482759935808X. Google Scholar

[17]

William Layton, "Introduction to the Numerical Analysis of Incompressible Viscous Flow,'', SIAM, (2008), 978. Google Scholar

[18]

O. P. Le Maître and O. M. Knio, "Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics,'', Springer, (2010), 978. Google Scholar

[19]

Catherine E. Powell and Howard C. Elman, Block-diagonal preconditioning for spectral stochastic finite-element systems,, IMA J. Numer. Anal., 29 (2009), 350. doi: 10.1093/imanum/drn014. Google Scholar

[20]

Catherine E. Powell and David Silvester, Preconditioning steady-state Navier-Stokes equations with random data,, MIMS Eprint 2012.35, (2012). Google Scholar

[21]

David Silvester, Howard Elman, David Kay and Andrew Wathen, Efficient preconditioning of the linearised Navier-Stokes equations for incompressible flow,, J. Comput. Appl. Math., 128 (2001), 261. doi: 10.1016/S0377-0427(00)00515-X. Google Scholar

[22]

David Silvester, Howard Elman and Alison Ramage, "Incompressible Flow and Iterative Solver Software (IFISS),'', Version 3.2, (2012). Google Scholar

[23]

J. C. Simo and F. Armero, Unconditional stability and long-term behaviour of transient algorithms for the incompressible Navier-Stokes and Euler equations,, Comput. Methods Appl. Mech. Engrg., 111 (1994), 111. doi: 10.1016/0045-7825(94)90042-6. Google Scholar

[24]

Dongbin Xiu, "Numerical Methods for Stochastic Computations: A Spectral Method Approach,'', Princeton University Press, (2010), 978. Google Scholar

show all references

References:
[1]

Uri M. Ascher, Steven J. Ruuth and Brian T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations,, SIAM J. Numer. Anal., 32 (1995), 797. doi: 10.1137/0732037. Google Scholar

[2]

Alexei Bespalov, Catherine E. Powell and David Silvester, A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data.,, MIMS Eprint 2011.91, (2011). Google Scholar

[3]

Jonathan Boyle, Milan Mihajlović and Jennifer Scott, HSL_MI20: an efficient {AMG} preconditioner for finite element problems in 3D,, Internat. J. Numer. Methods Engrg, 82 (2010), 64. Google Scholar

[4]

H. Damanik, J. Hron, A. Ouazzi and S. Turek, A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes,, J. Comput. Phys., 228 (2009), 3869. doi: 10.1016/j.jcp.2009.02.024. Google Scholar

[5]

Philip G. Drazin, "Introduction to Hydrodynamic Stability,'', Cambridge University Press, (2002). Google Scholar

[6]

Howard Elman, Milan Mihajlović and David Silvester, Fast iterative solvers for buoyancy driven flow problems,, J. Comput. Phys., 230 (2011), 3900. doi: 10.1016/j.jcp.2011.02.014. Google Scholar

[7]

Howard C. Elman, Alison Ramage and David J. Silvester, Algorithm 866: IFISS, a MATLAB toolbox for modelling incompressible flow,, ACM Trans. Math. Softw., 33 (2007), 2. Google Scholar

[8]

Howard C. Elman, David J. Silvester and Andrew J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations,, Numer. Math., 90 (2002), 665. doi: 10.1007/s002110100300. Google Scholar

[9]

Howard C. Elman, David J. Silvester and Andrew J. Wathen, "Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics,'', Oxford University Press, (2005). Google Scholar

[10]

Howard C. Elman and Ray S. Tuminaro, Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations,, Electron. Trans. Numer. Anal., 35 (2009), 257. Google Scholar

[11]

Oliver G. Ernst, Catherine E. Powell, David J. Silvester and Elisabeth Ullmann, Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data,, SIAM J. Sci. Comput., 31 (2009), 1424. doi: 10.1137/070705817. Google Scholar

[12]

P. M. Gresho, D. K. Gartling, J. R. Torczynski, K. A. Cliffe, K. H. Winters, T. J. Garratt, A. Spence and J. W. Goodrich, Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re=800 stable?,, Internat. J. Numer. Methods Fluids, 17 (1993), 501. doi: 10.1002/fld.1650170605. Google Scholar

[13]

Philip M. Gresho, David F. Griffiths and David J. Silvester, Adaptive time-stepping for incompressible flow; Part I: Scalar advection-diffusion,, SIAM J. Sci. Comput., 30 (2008), 2018. doi: 10.1137/070688018. Google Scholar

[14]

P. M. Gresho and R. L. Sani, "Incompressible Flow and the Finite Element Method: Volume 2: Isothermal Laminar Flow,'', John Wiley, (1998). Google Scholar

[15]

David A. Kay, Philip M. Gresho, David F. Griffiths and David J. Silvester, Adaptive time-stepping for incompressible flow; Part II: Navier-Stokes equations,, SIAM J. Sci. Comput., 32 (2010), 111. doi: 10.1137/080728032. Google Scholar

[16]

David Kay, Daniel Loghin and Andrew Wathen, A preconditioner for the steady-state Navier-Stokes equations,, SIAM J. Sci. Comput., 24 (2002), 237. doi: 10.1137/S106482759935808X. Google Scholar

[17]

William Layton, "Introduction to the Numerical Analysis of Incompressible Viscous Flow,'', SIAM, (2008), 978. Google Scholar

[18]

O. P. Le Maître and O. M. Knio, "Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics,'', Springer, (2010), 978. Google Scholar

[19]

Catherine E. Powell and Howard C. Elman, Block-diagonal preconditioning for spectral stochastic finite-element systems,, IMA J. Numer. Anal., 29 (2009), 350. doi: 10.1093/imanum/drn014. Google Scholar

[20]

Catherine E. Powell and David Silvester, Preconditioning steady-state Navier-Stokes equations with random data,, MIMS Eprint 2012.35, (2012). Google Scholar

[21]

David Silvester, Howard Elman, David Kay and Andrew Wathen, Efficient preconditioning of the linearised Navier-Stokes equations for incompressible flow,, J. Comput. Appl. Math., 128 (2001), 261. doi: 10.1016/S0377-0427(00)00515-X. Google Scholar

[22]

David Silvester, Howard Elman and Alison Ramage, "Incompressible Flow and Iterative Solver Software (IFISS),'', Version 3.2, (2012). Google Scholar

[23]

J. C. Simo and F. Armero, Unconditional stability and long-term behaviour of transient algorithms for the incompressible Navier-Stokes and Euler equations,, Comput. Methods Appl. Mech. Engrg., 111 (1994), 111. doi: 10.1016/0045-7825(94)90042-6. Google Scholar

[24]

Dongbin Xiu, "Numerical Methods for Stochastic Computations: A Spectral Method Approach,'', Princeton University Press, (2010), 978. Google Scholar

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