تجزیه و تحلیل عملکرد از شوارتز موازی قبل از تهویه در جریان های کانال آشفته LES

We present a comparative study of parallel Schwarz preconditioners in the solution of linear systems arising in a Large Eddy Simulation (LES) procedure for turbulent plane channel flows. This procedure applies a time-splitting technique to suitably filtered Navier–Stokes equations, in order to decouple the continuity and momentum equations, and uses a semi-implicit scheme for time integration and finite volumes for space discretisation. This approach requires the solution of four sparse linear systems at each time step, accounting for a large part of the overall simulation; hence the linear system solvers are a crucial component in the whole procedure. Several preconditioners are applied in the simulation of a reference test case for the LES community, using discretisation grids of different sizes, with the aim of analysing the effects of different algorithmic choices defining the preconditioners, and identifying the most effective ones for the selected problem. The preconditioners, coupled with the GMRES method, are run within SParC-LES, a recently developed LES code based on the PSBLAS and MLD2P4 libraries for parallel sparse matrix computations and preconditioning.

Large Eddy Simulation (LES) is a widely used approach for detailed study of turbulent flows in small/medium-scale applications. Although it has a lower computational cost than Direct Numerical Simulation (DNS), its application to realistic flows remains a computationally intensive procedure. In this work we focus on large and sparse linear systems arising in a LES procedure for turbulent wall-bounded flows, and analyse the performance of parallel Schwarz preconditioners in solving these systems by GMRES [1]. The linear systems stem from the application of a time-splitting technique to suitably filtered Navier–Stokes equations, to decouple the continuity and momentum equations, and from a finite-volume discretisation of the resulting equations (see the next section for details). Their solution accounts for a significant part of the overall computational effort; thus the use of efficient preconditioners is critical for the efficiency of the overall simulation. In our analysis we use the preconditioners implemented in the Multilevel Domain Decomposition Parallel Preconditioners Package based on PSBLAS (MLD2P4) [2], coupled with the GMRES solver from the Parallel Sparse BLAS (PSBLAS) library [3]. The solution of the systems is performed within SParC-LES [4], a parallel code recently developed for the simulation of turbulent channel flows, which is run on a test case used as a standard benchmark in the Italian LES community [5]. This study differs from previous work in [6] and [7] because it provides a more detailed analysis of several preconditioners for all the linear systems arising in the LES procedure, applied to a widely used reference test case. It is also worth noting that the results discussed in this paper guided our choice of the preconditioners in a complete simulation of the selected test case with the SParC-LES code (see [4]). The paper is organised as follows. In Section 2 we briefly outline the LES approach implemented in SParC-LES, with the aim of introducing the above-mentioned linear systems. In Section 3 we give a very brief overview of the preconditioners implemented in MLD2P4. Finally, in Section 4 we present our analysis of the preconditioners. We also provide some conclusions in Section 5

The work presented in this paper was devoted to analysing the performance of different parallel Schwarz preconditioners in the solution of linear systems arising in a LES procedure for turbulent channel flows. The preconditioners were applied within SParC-LES, a LES code recently developed by exploiting the PSBLAS and MLD2P4 libraries, implementing parallel sparse linear algebra kernels and preconditioners. We compared one-level and multilevel Schwarz preconditioners, with the final aim of achieving the best tradeoff among convergence, time and scalability objectives on a test case widely used by the LES community. The results show that, on the pressure system, the two-level and three-level preconditioners using a distributed block-Jacobi solver at the coarsest level provide a significant reduction of the number of iterations with respect to one-level RAS, which generally translates into smaller execution times and good scalability, especially on the largest grid. Conversely, the features of the velocity systems render pointless the use of sophisticated preconditioners in their solution. We are also confident that the above conclusions can be taken as guidelines for applying SParC-LES to test cases with increasing Reynolds numbers, since the velocity systems will still be well-conditioned, while the worse conditioning of the pressure system will render the multilevel preconditioners even more attractive and effective in a parallel computing setting. Finally, we note that, for all the three grids, the time spent in the linear system solvers accounts for approximately 60% of the total execution time, and is thus quite significant. When doing a test run using no preconditioner for the velocity systems and the very simple Jacobi preconditioner for the pressure system, the corresponding figure was about 90%; thus, the significant improvement in execution time gained by using an effective preconditioner for the pressure system also makes it quite important to have a good parallelisation of the rest of the application (see [4] for details).