Preconditioning in parallel for fractional step Navier--Stokes solvers

Abstract

Preconditioning for the Pressure Poisson Equation, used with the fractional step Navier-Stokes solvers, is studied. The Pressure Poisson Equation results from the segregated calculation of the velocity and pressure in the momentum equations, with the divergence of the velocity as the source term. The coefficient matrix of the Pressure Poisson Equation is dependent only upon grid-size, and thus preconditioners need to be constructed only once initially, and used for all subsequent time steps. Several preconditioning techniques are studied, including Jacobi, incomplete matrix decomposition variants, and sparse approximate inverses. The test case is a three dimensional turbulent channel flow, with the domain discretised using a structured nonstaggered grid. Parallel computing is performed on a cluster of processors by message passing, with domain partitioning to avoid the use of global gather Contents C20and scatter operations and to minimise the effect of communication bandwidth. The preconditioners are all constructed based on the local grid partition. For the sparse approximate inverse preconditioners, cell dependencies are bounded to limit communication across grid partition boundaries. The effect of a defined sparsity pattern is also investigated. The optimum sparsity pattern and the dependency on the neighbouring cells are found to be influenced by the grid ratios in each axis direction.

Parallel Speed-Up of Preconditioned Fractional Step Navier-Stokes Solvers

“…Those parameters were chosen from the optimum value for this case, which were obtained from preliminary tests. As the previous study [4] has shown that the optimum defined pattern depends on grid spacing or ratio, the pattern chosen here includes cell dependencies in the y direction only, omitting the dependencies in x and z directions. The SAI method includes 11 elements in each row of the preconditioner matrix, consisting of the diagonal element and 10 elements representing the dependencies to each 5-cells above and below the respective cell.…”

“…[3]. The effectiveness of preconditioners on the parallel pressure correction method for turbulent channel flow was studied in [4]. In that study, the sparse approximate inverse with defined sparsity pattern was found to have comparable performance to the adaptive approach.…”