An Efficient Direct Algorithm for the Solution of the Fourth-Order Poisson Equation on Loosely Coupled Parallel Computers

摘要

For the application of LES and DNS to a wide range of problems, including engineering applications, it is important to use efficient algorithms for the integration of time-dependent flows on loosely coupled parallel computers such as PC clusters. As it is well known, the main difficulty for the simulation of incompressible flows is the discrete Poisson equation. The Direct Schur-Fourier Decomposition (DSFD) method allows to solve each Poisson equation to almost machine accuracy using only one all-to-all communication. A previous version of this solver was used to solved the well-known three-dimensional (3D) second-order Poisson equation which has seven diagonals. If a fourth-order symmetry-preserving discretization by Verstappen and Veldman is used a fourth-order Poisson equation, which has nineteen diagonals, must be solve in order to satisfy exactly the subtle kinetic energy balance. Here, an overview of the fourth-order algorithm is given; its main advantages and limitations are discussed focusing on the main differences from its second-order counterpart. The DNS results of a turbulent natural convection problem, obtained using DSFD on a PC cluster is presented in a companion paper. Benchmark results illustrating the robustness and scalability of the method on a PC cluster with 100 Mbits/s network are presented and discussed.