INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2006; 50:579–596 Published online 7 September 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/d.1063 Multiple semi-coarsened multigrid method with application to large eddy simulation F. E. Ham 1; ‡ , F. S. Lien 2; ∗; † and A. B. Strong 2; § 1 Center for Integrated Turbulence Simulations; Stanford University; Stanford; CA 94305; U.S.A. 2 Department of Mechanical Engineering; University of Waterloo; University Avenue West; Waterloo; Ontario; Canada N2L 3G1 SUMMARY The Multiple Semi-coarsened Grid (MSG) multigrid method of Mulder (J. Comput. Phys. 1989; 83:303 –323) is developed as a solver for fully implicit discretizations of the time-dependent incom- pressible Navier–Stokes equations. The method is combined with the Symmetric Coupled Gauss–Seidel (SCGS) smoother of Vanka (Comput. Methods Appl. Mech. Eng. 1986; 55:321–338) and its robust- ness demonstrated by performing a number of large-eddy simulations, including bypass transition on a at plate and the turbulent thermally-driven cavity ow. The method is consistently able to reduce the non-linear residual by 5 orders of magnitude in 40–80 work units for problems with signicant and varying coecient anisotropy. Some discussion of the parallel implementation of the method is also included. Copyright ? 2005 John Wiley & Sons, Ltd. KEY WORDS: MSG multigrid; symmetric coupled Gauss–Seidel; LES; incompressible Navier–Stokes 1. INTRODUCTION Implicit discretizations of the Navier–Stokes equations are known to have favourable prop- erties relative to their explicit counterparts. The most obvious is the relaxation or removal of the numerical stability constraint on the computational time step. With respect to large eddy simulation (LES), where a broad range of spatial and temporal scales must be accu- rately resolved, there are a number of other more subtle benets. For example, the implicit discretization of the incompressible or variable-density form of the Navier–Stokes equations results in an operator symmetry in space and time that can have very favourable kinetic ∗ Correspondence to: F. S. Lien, Department of Mechanical Engineering, University of Waterloo, University Avenue West, Waterloo, Ontario, Canada N2L 3G1. † E-mail: fslien@uwaterloo.ca ‡ E-mail: fham@stanford.edu § E-mail: astrong@sunwise.uwaterloo.ca Contract=grant sponsor: Natural Sciences and Engineering Research Council of Canada (NSERC) Received 19 January 2005 Revised 11 July 2005 Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 11 July 2005