Preconditioned iterative methods for coupled discretizations of fluid flow problems

Computational fluid dynamics, where simulations require large computation times, is one of the areas of application of high performance computing. Schemes such as the SIMPLE (semi-implicit method for pressure-linked equations) algorithm are often used to solve the discrete Navier-Stokes equations. Generally these schemes take a short time per iteration but require a large number of iterations. For simple geometries (or coarser grids) the overall CPU time is small. However, for finer grids or more complex geometries the increase in the number of iterations may be a drawback and the decoupling of the differential equations involved implies a slow convergence of rotationally dominated problems that can be very time consuming for realistic applications. So we analyze here another approach, DIRECTO, that solves the equations in a coupled way. With recent advances in hardware technology and software design, it became possible to solve coupled Navier-Stokes systems, which are more robust but imply increasing computational requirements (both in terms of memory and CPU time). Two approaches are described here (band block LU factorization and preconditioned GMRES) for the linear solver required by the DIRECTO algorithm that solves the fluid flow equations as a coupled system. Comparisons of the effectiveness of incomplete factorization preconditioners applied to the GMRES (generalized minimum residual) method are shown. Some numerical results are presented showing that it is possible to minimize considerably the CPU time of the coupled approach so that it can be faster than the decoupled one.