Abstract

Linear solvers form the inner core of many simulations in science and engineering. For example, in computational fluid dynamics (CFD), the numerical solution of the Navier-Stokes equations require the solution of a large, sparse linear system to obtain the pressure solution and, in consequence, ensure the consistency of the velocity field. While the discretization of the partial differential equations involved is a local operation and thus can be parallelized easily, the linear solution is a global operation and thus imposes a significant challenge for the overall scalability of the simulation. Algebraic multigrid (AMG) methods provide optimal iterative linear solvers for a wide class of problems, i.e. they show a linear scaling with respect to N in both compute time and memory requirements. They automatically construct a hierarchy of linear systems to adequately deal with the different frequencies of the underlying problem. To this end, a so-called setup phase is carried out before the iteration starts. In this talk, we present the parallel scalability of the algebraic multigrid solver SAMG inside an OpenFOAM CFD solver. We focus on how to improve the speed-up (i.e. strong scaling) for large-scale simulations (10-100Ms of degrees of freedom, 10000s of time steps) on industrially relevant processor numbers. To improve the parallel performance, we combine several components: First, we employ parallel AMG algorithms that lower the communication overhead and thus improve the scalability[1,2]. Second, we only solve as accurate as needed: We reduce the number of iterations where possible, and use single precision arithmetic if the accuracy required allows so. Furthermore, we have developed a steering mechanism that carefully inspects how often the AMG setup phase needs to be carried out. Finally, for selected examples we have carried out a compilation optimization process to gain further speed-up. [1] H. De Sterck, U.M. Yang, and J.J. Heys, Reducing Complexity in Parallel Algebraic Multigrid Preconditioners, SIAM J. on Matrix Analysis and Applications, 27 (2006), pp. 1019-1039. UCRL-JRNL-206780. [2] R.D. Falgout and J.B. Schroder, Non-Galerkin Coarse Grids for Algebraic Multigrid,SIAM J. Sci. Comput., 36 (2014), pp. C309-C334. LLNL-JRNL-641635.