Efficient Parallel AMG Methods for Approximate Solutions of Linear Systems in CFD Applications

Abstract

In engineering practice quite simple algebraic multigrid (AMG) methods with inexpensive coarse-grid selection and constant interpolation are frequently found to be the method of choice to solve systems of linear equations where accurate solutions of these systems are not needed.In a case study we analyze the poor parallel performance of such algorithms on common multicore cluster architectures and suggest using a class of algorithms with hybrid coarse-grid selection featuring a superior parallel performance.

Krylov‐accelerated algebraic multigrid for semi‐definite and nonsymmetric systems in computational fluid dynamics

“…For calculations with more than four parallel processes, the diagrams in Figure show the computing time of two different ways to distribute the processes to the nodes: if at most four processes run on one node, the computing time decreases with increasing number of parallel processes in a relatively uniform manner (a detailed discussion of the parallel performance of AMG in such a configuration can be found in Emans ); if all available eight cores of the cluster nodes are used, the same calculation takes significantly more time, for example, 242.8 s compared with 152.8 s for problem S3 solved with SA‐V‐CG. This is because of the limited on‐chip memory bandwidth.…”

Coarse-grid treatment in parallel AMG for coupled systems in CFD applications

“…Trottenberg et al [10] describe the principal options: either the distributed coarse-grid system is solved in parallel or, at some stage, the system is combined on one of the processors, eventually additional coarse-grids are constructed and system of the coarsest grid is solved directly by one processor. As far as we know, a systematic examination of the practical consequences of the choice of the coarse-grid treatment strategy has not been published yet; according to our own experience, the latter method, referred to as agglomeration in this paper, is appropriate for scalar systems, see Emans [3]. It could be demonstrated that AMG with agglomeration has good convergence properties and shows good parallel efficiency.…”

“…A generalised method to compute the optimal relaxation parameter was devised by Yang [11] for symmetric positive definite systems, but, to our knowledge, it did not attain practical relevance since it is too expensive. In our own practical experience poor quality of the coarse-grid solution by the iterative solver can lead to significant deterioration of the convergence rates compared to the serial calculation with the same multigrid method, see Emans [3], in particular if schemes with rapid coarsening such as Smoothed Aggregation are used. The described scheme where the linear system associated with the coarsest grid is solved in parallel by a block-Jacobi scheme is referred to iterative coarse-grid solver; we will use the short IT for this scheme in this paper.…”

“…aggregates are not intersected by inter-domain boundaries and the smoothing is done only between the interior points of each domain. More details of the parallel implementation have been discussed in a previous publication by the author [3]. This algorithm is used as a preconditioner of a GMRES method.…”

“…the matrix of the pressure-correction equation for both mentioned methods is symmetric positive definite in many cases. Linear solvers based on algebraic multigrid (AMG) are among the most efficient methods to solve the systems in the context of industrial software, see Emans [3]. The price to to be paid for this decoupling of the equations is usually a slower convergence of the outer algorithm, e.g.…”