Keywords: computational fluid dynamics, large sparse systems, non-symmetric matrices, preconditioning, Krylov subspace methods, additive Schwarz, incomplete LU, ill-conditioned factors, grid-point ordering, PETSc.

coupled system of partial differential equations (Reynolds-Averaged Navier-Stokes) 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 non-symmetric, particularly when the stationary flow on which the derivation is based, is dominated by advection and a centered spatial-discretization 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 three-dimensional 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 grid-block approach: the sparse Jacobian matrix consisting in a collection of dense m-by-m matrices, when there are m flow variables per point (a grid-point 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 grid-block ILU factorization. We note here that for reasons regarding CPU time cost and difficulties to extend point-based algorithms to block ones, only static-pattern 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 ill-conditioning. 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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