Parallel Implementation of Curvilinear High-order Formulas

High-order formulas are required for differencing and filtering the Navier-Stokes equations in order to obtain the needed accuracy for a variety of CFD applications. The parallel performance issue relevant to one of these methods, the compact scheme, is studied in this paper with emphasis on the associated implicit operators. Three procedures were selected: the one-sided method, the parallel diagonal dominant (PDD) method, and the parallel Thomas algorithm (PTA) method. These parallel procedures were implemented in the AFRL code, FDL3DI. Kernel codes were also developed to extract some inherent performance features of these methods. Some of the calculations combine the methods for compact differencing and filtering. In general, the procedures based on the one-sided schemes produced accurate results and good parallel performance (efficiency, speedup, and scalability). The procedures that combine the one-sided schemes and PDD also performed well. However, parallel calculations that use the PDD method for both compact differencing and filtering produced the wrong results for low-order filters. On the other hand, high-order filters cause PDD to be very expensive. The one-sided method leads to super-scalable calculations when the number of processors is low. For PDD, increasing the number of grid points in the derivative-difference direction leads to better speedup, as does an increase in the number of right-hand side (RHS) columns. In standard implementation (i.e. without engaging the processors during the idle time), the PTA procedure has a very poor parallel performance in comparison to PDD and the one-sided formulations. However, the procedure tends to be more accurate.