Preconditioning in parallel for fractional step Navier--Stokes solvers

Abstract

Preconditioning for the Pressure Poisson Equation, used with the fractional step Navier--Stokes solvers, is studied. The Pressure Poisson Equation results from the segregated calculation of the velocity and pressure in the momentum equations, with the divergence of the velocity as the source term. The coefficient matrix of the Pressure Poisson Equation is dependent only upon grid-size, and thus preconditioners need to be constructed only once initially, and used for all subsequent time steps. Several preconditioning techniques are studied, including Jacobi, incomplete matrix decomposition variants, and sparse approximate inverses. The test case is a three dimensional turbulent channel flow, with the domain discretised using a structured nonstaggered grid. Parallel computing is performed on a cluster of processors by message passing, with domain partitioning to avoid the use of global gather and scatter operations and to minimise the effect of communication bandwidth. The preconditioners are all constructed based on the local grid partition. For the sparse approximate inverse preconditioners, cell dependencies are bounded to limit communication across grid partition boundaries. The effect of a defined sparsity pattern is also investigated. The optimum sparsity pattern and the dependency on the neighbouring cells are found to be influenced by the grid ratios in each axis direction. References M. Benzi. Preconditioning techniques for large linear systems: A survey. J. of Comp. Physics , 182(2):418--477, 2002. doi:10.1006/jcph.2002.7176 Y. Saad. Iterative Methods for Sparse Linear Systems . SIAM, Philadelphia, PA, USA, 2nd edition, 2003. V. S. Djanali, S. W. Armfield, and M. P. Kirkpatrick. Comparison of approximate inverse preconditioners for the fractional step Navier--Stokes equations. ANZIAM J. , 52:C581--C595, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3890 M. Ament, G. Knittel, D. Weiskopf, and W. Strasser. A parallel preconditioned conjugate gradient solver for the Poisson problem on a multi-gpu platform. Parallel, Distributed and Network-Based Processing (PDP), 2010 18th Euromicro International Conference , 583--592, 2010. doi:10.1109/PDP.2010.51 S.M. Xu, H.X. Lin, K. Wang, and W. Xue. Utilizing cuda for preconditioned gmres solvers. Proc. 8th Int‚Aol Symp. on Distributed Computing and Applications to Business, Engineering and Sciences (DCABES 2009) . S. Armfield and R. Street. An analysis and comparison of the time accuracy of fractional‚aistep methods for the Navier--Stokes equations on staggered grid. Int. J. Numer. Methods Fluids , 38:255--282, 2002. doi:10.1002/fld.217 H. L. Stone. Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Numer. Anal. , 5(3):530--558, 1968. doi:10.1137/0705044 G. E. Schneider and M. Zedan. A Modified strongly implicit procedure for the numerical solution of field problems. Numer. Heat Transfer , 4(1):1--19, 1981. doi:10.1080/01495728108961775 M. J. Grote and T. Huckle. Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. , 18(3):838--853, May 1997. doi:10.1137/S1064827594276552 M. Benzi and M. Tuma. A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. , 19(3):968--994, 1998. doi:10.1137/S1064827595294691 R. D. Moser, J. Kim, and N. N. Mansour. {Direct numerical simulation of turbulent channel flow up to Re$_\tau $ = 590}. Phys. Fluids , 11(4):943--945, 1999. doi:10.1063/1.869966