High-order accurate solution of the incompressible Navier–Stokes equations on massively parallel computers

The emergence of “petascale” supercomputers requires us to develop today’s simulation codes for (incompressible) flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power. In that spirit, we present a massively parallel high-order Navier–Stokes solver for large incompressible flow problems in three dimensions. The governing equations are discretized with finite differences in space and a semi-implicit time integration scheme. This discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers. The iterative solver for the pressure uses a highly efficient commutation-based preconditioner which is robust with respect to grid stretching. The efficiency of the implementation is further enhanced by carefully setting the (adaptive) termination criteria for the different iterative solvers. The computational work is distributed to different processing units by a geometric data decomposition in all three dimensions. This decomposition scheme ensures a low communication overhead and excellent scaling capabilities. The discretization is thoroughly validated. First, we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow. Second, we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria. Third, Orr–Sommerfeld and Squire eigenmodes for plane Poiseuille flow are simulated and compared to analytical results. Fourth, the practical applicability of the implementation is tested for transitional and turbulent channel flow. The results are compared to solutions from a pseudospectral solver. Subsequently, the performance of the commutation-based preconditioner for the pressure iteration is demonstrated. Finally, the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to O(104)O(104) processing units and O(1011)O(1011) grid points.