Design of a Parallel Hybrid Direct/Iterative Solver for CFD Problems

Abstract

Abstract-We discuss the parallel implementation of a hybrid direct/iterative solver for a special class of saddle point matrices arising from the discretization of the steady Navier-Stokes equations on an Arakawa C-grid, the F -matrices.The two-level method described here has the following properties: (i) it is very robust, even hat comparatively high Reynolds Numbers; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively. The implementation focusses on generality, modularity, code reuse and recursiveness. The solver is implemented using building blocks of the Trilinos libraries. We show its performance on a parallel computer for the NavierStokes equations.

A staggered‐grid multilevel incomplete LU for steady incompressible flows

“…Rather than reviewing the method and theory in detail, we only briefly present it here. For details, see Wubs and Thies 24 and Thies and Wubs 25 …”

“…In this article, we present a novel multilevel preconditioning method which is specially designed for the 3D Navier‐Stokes equations. In Section 2, we first describe the two‐level ILU preconditioner as introduced in Wubs and Thies 24 and Thies and Wubs 25 . After this, we generalize the two‐level method to a multilevel method in Section 3.…”

A fine-grained block ILU scheme on regular structures for GPGPUs

“…One common practice to extract parallelism is level scheduling, such as wavefront ordering, but its benefit is limited by synchronization overhead and still insufficient parallelism. Other techniques such as coloring or graph partitioning allow efficient coarse-grained to mid-grained parallelism with little overhead (see, for example [2][3][4][5][6]), at the expense of slightly reduced mathematical efficiency. More detailed reviews can be found in [7][8][9].…”

Accelerating Poisson solvers in front tracking method using parallel direct methods

“…Iterative solvers are highly memory efficient and easier to be parallelized on the OpenMP or shared memory platforms [3]. Furthermore, iteration methods often require custom numerical tools such as preconditioning techniques to be efficient [4]. In addition, in problems involving highly ill-conditioned matrices, iterative solvers suffer from well performance.…”