Vectorization of Algorithms in Computational Fluid Dynamics on the CRAY-1 Vector Computer

Abstract

Among computational fluid dynamicists the tremendous impact of MacCormack’s contributions to the development of computational tools for integrating the governing equations for inviscid and viscous flows is well recognized, and this is documented by numerous publications based on the use of his methods for integrating steady as well as time-dependent Euler and Navier-Stokes equations. Best known are MacCormack’s purely explicit predictor-corrector versions |1,2| of the two-level scheme of Lax and Wendroff. Explicit schemes are easily applied, and can, in general, be completely vectorized, but they have to satisfy severe stability conditions with respect to the marching step size. Such conditions are most stringent for viscous flows if the wall-normal, boundary-layer direction is to be resolved properly, with step sizes for turbulent flows of the order of 0.00005 times the characteristic length involved. For the explicit method to be stable in time-dependent calculations, the time-wise step must then be chosen proportional to the square of the corresponding spatial step size divided by the kinematic viscosity. In the case of inviscid flow described by the Euler equations, the time step size has to be proportional only to the spatial step size itself divided by an appropriate velocity, this being the Courant-Friedrichs-Lewy condition. To overcome these restrictions, at least partially, MacCormack introduced symmetric operator splitting (i.e. the concept of breaking up the multi-dimensional operator into sequences of one-dimensional operators without factorization) such that one large step is performed for the surface tangential direction, while for the crucial wallnormal direction that time step is divided into many single steps so that all steps satisfy the stability conditions |2,5,6|. While the Euler equations in three dimensions |4,5| were integrated successfully and in CPU times which were reasonable for serial computers, this could not be done for the Navier-Stokes equations. Even in two dimensions, in particular for high-Reynolds number transonic flows |6|, the solutions were very time consuming.