A direct Schur–Fourier decomposition for the efficient solution of high‐order Poisson equations on loosely coupled parallel computers

Abstract

SUMMARYIn this paper a parallel direct Schur-Fourier decomposition (DSFD) algorithm for the direct solution of arbitrary order discrete Poisson equations on parallel computers is proposed. It is based on a combination of a Direct Schur method and a Fourier decomposition and allows to solve each Poisson equation almost to machine accuracy using only one communication episode. Thus, it is well suited for loosely coupled parallel computers, that have a high network latency compared with the CPU performance. Several three-dimensional direct numerical simulations (DNS) of wall-bounded turbulent incompressible ows have been carried out using the DSFD algorithm. Numerical examples illustrating the robustness and scalability of the method on a PC cluster with a conventional 100 Mbits=s network are also presented.

Numerical analysis of conservative unstructured discretisations for low Mach flows

“…In both predictor and corrector substeps, the pressure Poisson equation is solved using a direct Schur decomposition [34,35]. Other variants for the fractional step method applied to low Mach flows consider a velocity projection, which results in a variable coefficient Poisson equation [13,14].…”

A scalable parallel Poisson solver for three-dimensional problems with one periodic direction

“…However, the DSFD algorithm that is very efficient on PC clusters cannot be used for an arbitrarily large number of processors and problem size, mainly due to the RAM memory requirements [1] and the size of communications that grow fast with the number of processors and mesh sizes. These problems limit the DSFD solver scalability especially for high-order numerical schemes making it not applicable for large-scale problems on supercomputers using hundreds (or thousands) of processors.…”

“…In contrast, when using supercomputers the number of CPUs is far greater and network performance much better (especially in terms of latency). It is important to note that depending on the computer architecture, the number of processors and the scale of the problem being solved, these considerations may become even more important than the arithmetical complexity of the algorithm [1]. Therefore algorithms that work well on supercomputers may not work efficiently on PC clusters due to the poor network performance and vice versa.…”

“…Our previous version of the code [1,8,9] was conceived for lowcost PC clusters with poor network performance. In that context, a direct Schur complement based method was a good choice for the parallel solution of this set of 2D problems because systems arising from non-uniform meshes can be solved without increasing the computational costs.…”

“…Furthermore, since the parallelisation of FFT is not efficient on loosely-coupled parallel computers parallelisation in the periodic direction was discarded. Such parallel Poisson solver, named Direct Schur-Fourier Decomposition (DSFD) algorithm [1,9], has been successfully used to perform DNS simulations [4,5] of turbulent natural convection flows in enclosed cavities.…”