Sources of Parallel Inefficiency for Incompressible CFD Simulations (Research Note)

Parallel multigrid methods are very prominent tools for solving huge systems of (non-)linear equations arising from the discretisation of PDEs, as for instance in Computational Fluid Dynamics (CFD). The superiority of multigrid methods in regard of numerical complexity mainly stands and falls with the smoothing algorithms (‘smoother’) used. Since the inherent highly recursive character of many global smoothers (SOR, ILU) often impedes a direct parallelisation, the application of block smoothers is an alternative. However, due to the weakened recursive character, the resulting parallel efficiency may decrease in comparison to the sequential performance, due to a weaker total numerical efficiency. Within this paper, we show the consequences of such a strategy for the resulting total efficiency if incorporated into a parallel CFD solver for 3D incompressible flow. Moreover, we compare this parallel version with the related optimised sequential code in FeatFlow and we analyse the numerical losses of parallel efficiency due to communication costs, numerical efficiency and finally the choice of programming language (C++ vs. F77). Altogether, we obtain quite surprising, but more realistic estimates for the total efficiency of such a parallel CFD tool in comparison to the related ‘optimal’ sequential version. 1 Numerical and Algorithmic Approach A parallel 3D code for the solution of the incompressible nonstationary NavierStokes equations ut − ν∆u+ (u · ∇)u+∇p = f , ∇ · u = 0 (1) has been developed. This code is an adaptation of the existing sequential FeatFlow solver (see www.featflow.de). For a detailed description of the numerical methods applied see [1, 3]. Here we restrict ourselves to a very brief summary of the mathematical background. Equation (1) is discretised separately in space and time. First, it is discretised in time by one of the usual second order methods known from the treatment of ordinary differential equations (Fractional-Step-θscheme, Crank-Nicolson-scheme). Space discretisation is performed by applying