Numerical testing of a data-parallel LU relaxation method for compressible DNS

Abstract

Introduction A high-order conservative hybrid finite difference scheme was developed for solving compressible turbulent boundary layer DNS problems. The scheme is hybrid in the sense that it contains both high-order central differencing to resolve the large range of length scales that occur in compressible turbulent flow and upwind biased flux vector splitting to provide necessary high wavenumber dissipation. The hybrid scheme, ncfvs (non-compact flux vector splitting), compares favorably with high-order compact schemes in numerical tests but does not contain the costly tridiagonal inversion that is found in compact schemes. When coupled to third-order Runge-Kutta time integration, this hybrid scheme is restricted by stability to a CFL = 0.6. This results in a very small time step due to the fine grid spacing necessary to resolve the length scales occurring in high Reynolds number flows. Although the hybrid scheme is fully parallelizable and optimized to run efficiently on the 512 node partition of the CM-5, a massively parallel supercomputer, solutions may take days due to this small time step restriction. If the time step could be increased by using an implicit method, statistically steady solutions could be found in a much shorter time. Rai, Gatski, and Erlebacher used an approximate factorization method to solve the implicit problem. Although effective, this method is difficult to efficiently implement on a parallel machine because of the complicated communication paths required in the inversion of tridiagonal systems. In this paper, we propose a high-order accurate formulation of the Data Parallel Lower-Upper Relaxation method (DP-LUR). The DP-LUR method has been used to efficiently solve many steady-state problems and is completely parallelizable. We modify this method to include secondorder time accuracy and then we compare the results of a compressible turbulent boundary layer problem to the third-order Runge-Kutta results of the same problem.