A Direct Parallel Algorithm for the Efficient Solution of the Pressure-Correction Equation of Incompressible Flow Problems Using Loosely Coupled Computers

Abstract

The numerical simulation of time-accurated complex ows needs large computational resources. For the case of incompressible ows, the solution of the pressure-correction equation is typically the main bottleneck, especially on loosely coupled parallel computers such as PC clusters. An algorithm intended to solve this problem is presented. It is a variant of the Schur complement method that uses direct solvers for each subdomain and for the interface equation. The inverse of the interface matrix is evaluated and stored in parallel. Simulation of turbulent natural convection is used as a benchmark to show its potential and limitations.

Direct numerical simulation of a NACA0012 in full stall

“…This can be circumvented by using a Fast Fourier Transform (FFT) based algorithm (Swarztrauber, 1977;Soria et al, 2002).…”

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

“…It is based on a combination of a direct Schur [11,12] method and a Fourier decomposition [8]. Fourier decomposition allows to decouple the unknowns in the periodic direction.…”

“…Each of the n p1 system of linear equations (46) is solved with a direct Schur decomposition (DSD) method [12,13]. To simplify the notation, the hats and sub-indices are dropped and each of the n p1 block diagonal equations (46) to be solved is denoted as…”

“…Only one all-to-all communication episode is needed to solve each 3D Poisson equation to machine accuracy. Previous versions of this algorithm [12,13] were used to solve the second-order Poisson equation. Now, a general extension of the algorithm for higher order discrete Poisson equations is presented.…”

“…The Schur complement matrix only needs to be calculated in a pre-processing stage and it allows to solve each Poisson equation almost to machine accuracy using only one all-to-all communication episode [12].…”