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CivilComp Proceedings
ISSN 17593433 CCP: 95
PROCEEDINGS OF THE SECOND INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED, GRID AND CLOUD COMPUTING FOR ENGINEERING Edited by: P. Iványi and B.H.V. Topping
Paper 39
A Study of ILU Factorization for Schwarz Preconditioners with Application to Computational Fluid Dynamics F. Pacull^{1}, S. Aubert^{1} and M. Buisson^{2}
^{1}Fluorem, Ecully, France
F. Pacull, S. Aubert, M. Buisson, "A Study of ILU Factorization for Schwarz Preconditioners with Application to Computational Fluid Dynamics", in P. Iványi, B.H.V. Topping, (Editors), "Proceedings of the Second International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering", CivilComp Press, Stirlingshire, UK, Paper 39, 2011. doi:10.4203/ccp.95.39
Keywords: computational fluid dynamics, large sparse systems, nonsymmetric matrices, preconditioning, Krylov subspace methods, additive Schwarz, incomplete LU, illconditioned factors, gridpoint ordering, PETSc.
Summary
coupled system of partial differential equations (ReynoldsAveraged
NavierStokes) is differentiated at a steady state equilibrium, in order to form
a Jacobian matrix with respect to the fluid variables discretized over the
computational domain. Typically, the resulting sparse linear matrix is
indefinite and nonsymmetric, particularly when the stationary flow on which the
derivation is based, is dominated by advection and a centered
spatialdiscretization scheme is used. If direct methods are known to be robust
for such matrices lacking most of the nice ordinary properties such as diagonal
dominance, the ever growing number of domain grid points used in
threedimensional industrial applications and the resulting matrix size and
bandwidth proscribe their use because of the induced memory requirements: until
now, preconditioned Krylov subspace solvers are inevitably employed for this
type of problem. Iterative methods being beyond the scope of this paper, we
focus here on classical preconditioners for CFD matrices and more precisely on
the additive Schwarz (AS) technique, an appropriate domain decomposition method
for parallel computers: the domain is split so that processes deal
simultaneously with independent subdomain problems. If incomplete LU (ILU)
factorization methods do not possess a high degree of parallelism, their use on
large sparse matrices is limited to a sequential task with the AS preconditioner
since the incomplete factors are only applied within the subdomains as local
operators. Also, the coupling between the flow variables at each point being
strong, ILU factorization can be advantageously associated with a gridblock
approach: the sparse Jacobian matrix consisting in a collection of dense
mbym matrices, when there are m flow variables per point (a gridpoint
based ordering is used, labeling the variables contiguously at each grid
point), operations on an upper level than grid blocks are treated as scalar
operations while lower operations involving block entries are optimized. In
addition the quotient graph, based on the grid, is used in the partitioning and
reordering processes. Eventually, a basic but efficient preconditioner for CFD
matrices is constructed by combining the AS method with a local gridblock ILU
factorization. We note here that for reasons regarding CPU time cost and
difficulties to extend pointbased algorithms to block ones, only staticpattern
ILU without pivoting is considered in the following. This preconditioned
iterative solver is the basis of our study, and we mention here that it has
been implemented with the portable, extensible toolkit for scientific
computations (PETSc) library from the Argonne National Laboratory (Argonne, IL,
USA). The drawback of this solver, when applied to CFD matrices, is a lack of
robustness: the preconditioner operator may strongly deviate from identity and
the incomplete factors may suffer from illconditioning. It was found that this
behavior may depend on the subdomain size order, the partitioning, the reording
strategy, the level of fill or some other parameters: to our knowledge, it is
rather unpredictable and in some degrees, not understood. In this paper we give at
first an overview of the literature, then describe the link between the
preconditioner efficiency and the ILU accuracy and instability of the triangular
solves within the subdomains, examine a few CFD cases for which the
preconditioner behaves poorly, and finally describe some preconditioner quality
estimators.
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