Fast iterative solvers for buoyancy driven flow problems

We outline a new class of robust and efficient methods for solving the Navier-Stokes equations with a Boussinesq model for buoyancy driven flow. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the efficiency of the chosen preconditioning schemes with respect to the discretization parameters.

[1]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[2]  R. Sani,et al.  Incompressible Flow and the Finite Element Method, Volume 1, Advection-Diffusion and Isothermal Laminar Flow , 1998 .

[3]  F. Wubs.  Notes on numerical fluid mechanics , 1985 .

[4]  Jaroslav Hron,et al.  A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes , 2009, J. Comput. Phys..

[5]  C. Vuik,et al.  SIMPLE‐type preconditioners for the Oseen problem , 2009 .

[6]  J. Szmelter.  Incompressible flow and the finite element method , 2001 .

[7]  K. Ragsdell,et al.  The energy method , 1975 .

[8]  Wulf G. Dettmer,et al.  An analysis of the time integration algorithms for the finite element solutions of incompressible Navier-Stokes equations based on a stabilised formulation , 2003 .

[9]  ElmanHoward,et al.  Fast iterative solvers for buoyancy driven flow problems , 2011 .

[10]  J. C. Simo,et al.  Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations , 1994 .

[11]  Dominic Davis,et al.  An operator-splitting finite-element approach to the 8:1 thermal-cavity problem , 2002 .

[12]  Brian Straughan,et al.  The Energy Method, Stability, and Nonlinear Convection , 1991 .

[13]  Rama Govindarajan,et al.  An Introduction to Hydrodynamic Stability , 2010 .

[14]  Pinhas Z. Bar-Yoseph,et al.  Stability of multiple steady states of convection in laterally heated cavities , 1999, Journal of Fluid Mechanics.

[15]  William Layton,et al.  Introduction to the Numerical Analysis of Incompressible Viscous Flows , 2008 .

[16]  H. Ninokata,et al.  Numerical Simulation of Oscillatory Convection in Low Prandtl Number Fluids Using Aqua Code , 1990 .

[17]  D FalgoutRobert.  An Introduction to Algebraic Multigrid , 2006 .

[18]  Mark A. Christon,et al.  Computational predictability of time‐dependent natural convection flows in enclosures (including a benchmark solution) , 2002 .

[19]  Anne C. Skeldon,et al.  Convection in a low Prandtl number fluid , 1996 .

[20]  Christopher Smethurst.  A Finite Element Solution of the Natural Convection Problem in 3D , 2010 .

[21]  R. L. Sani,et al.  Isothermal laminar flow , 2000 .

[22]  Horst Leipholz,et al.  The Energy Method , 1987 .

[23]  J. Scott,et al.  HSL MI 20 : An efficient AMG preconditioner for finite element problems in 3 D , 2010 .

[24]  Shihe Xin,et al.  An extended Chebyshev pseudo‐spectral benchmark for the 8:1 differentially heated cavity , 2002 .

[25]  David F. Griffiths,et al.  Adaptive Time-Stepping for Incompressible Flow Part II: Navier--Stokes Equations , 2010, SIAM J. Sci. Comput..

[26]  J. Scott,et al.  HSL_MI20: An efficient AMG preconditioner for finite element problems in 3D , 2010 .

[27]  Howard C. Elman,et al.  BOUNDARY CONDITIONS IN APPROXIMATE COMMUTATOR PRECONDITIONERS FOR THE NAVIER-STOKES EQUATIONS ∗ , 2009 .

[28]  K. H. Winters.  Oscillatory convection in liquid metals in a horizontal temperature gradient , 1988 .

[29]  R.D. Falgout,et al.  An Introduction to Algebraic Multigrid Computing , 2006, Computing in Science & Engineering.

[30]  David F. Griffiths,et al.  Adaptive Time-Stepping for Incompressible Flow Part I: Scalar Advection-Diffusion , 2008, SIAM J. Sci. Comput..