Implicit block data-parallel relaxation scheme of Navier-Stokes equations using graphics processing units

The granularity of computational fluid dynamics (CFD) generally refers to the point granularity parallelization as a unit of the grid when graphics processing units (GPUs) are utilized as the computing carrier. In commonly deployed implicit time advancement schemes, the parallel dimensionality must be reduced, resulting in the time advancement procedure becoming the only highly time-consuming step in the whole CFD computing procedures. In this paper, a block data-parallel lower-upper relaxation (BDPLUR) scheme based on Jacobi iteration and Roe's flux scheme is proposed and then implemented on a GPU. Numerical experiments are carried out and show that the convergence speed of the BDPLUR scheme, especially when implemented on a GPU, is approximately 10 times higher than that of the original data-parallel lower-upper relaxation (DPLUR) scheme and more than 100 times higher than that of the lower-upper symmetric Gauss‒Seidel (LUSGS) scheme. Moreover, the influence of different Courant-Friedrichs-Lewy (CFL) numbers on the convergence time is discussed, and different viscous matrices are compared. Standard cases are adopted to verify the effectiveness of the BDPLUR scheme.