Speeding up an unsteady flow simulation by adaptive BDDC and Krylov subspace recycling

Abstract: We deal with accelerating the solution of a sequence of large linear systems solved by an iterative Krylov subspace method. The sequence originates from time-stepping within a simulation of an unsteady incompressible flow. We apply a pressure correction scheme, and we focus on the solution of the Poisson problem for the pressure corrector. Its scalable solution presents the main computational challenge in many applications. The right-hand side of the problem changes in each time step, while the system matrix is constant and symmetric positive definite. The acceleration techniques are studied on a particular problem of flow around a unit sphere. Our baseline approach is based on a parallel solution of each problem in the sequence by nonoverlapping domain decomposition method. The interface problem is solved by the preconditioned conjugate gradient (PCG) method with the three-level BDDC preconditioner. Three techniques for accelerating the solution are gradually added to the baseline approach. First, the stopping criterion for the PCG iterations is studied. Next, deflation is used within the conjugate gradient method with several approaches to Krylov subspace recycling. Finally, we add the adaptive selection of the coarse space within the three-level BDDC method. The paper is rich in experiments with careful measurements of computational times on a parallel supercomputer. The combination of the acceleration techniques eventually leads to saving about one half of the computational time.