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

TL;DR

The paper tackles speeding up solving a sequence of linear systems with a constant matrix and varying RHS arising from the Poisson pressure solve in an unsteady incompressible flow. The authors build a pipeline combining a baseline three-level BDDC preconditioner, a normalization-based stopping criterion, Krylov subspace recycling with deflated PCG (leveraging the largest Ritz vectors), and adaptive enrichment of the BDDC coarse space via local eigenproblems. They demonstrate substantial time savings, including over 50% in transient Re = 100 regimes and over 40% for large-scale problems, with joint recycling and adaptivity delivering the strongest gains; results include careful timing on a parallel supercomputer and robust performance across parameter ranges. The work provides a practical, scalable acceleration framework for time-dependent incompressible flow simulations, showing that combining these established techniques yields synergistic benefits beyond what each achieves individually.

Abstract

We deal with accelerating the solution of a sequence of large linear systems solved by preconditioned conjugate gradient method (PCG). The sequence originates from time-stepping within a simulation of an unsteady incompressible flow. We apply a pressure correction scheme and 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 representative 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 PCG with the three-level BDDC preconditioner. As a preliminary step, an appropriate stopping criterion for the PCG iterations is chosen. Next, two techniques for accelerating the solution are gradually added to the baseline approach. Deflation is used within PCG 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 more than 40 % of the computational time.

Figures (6)

• Figure 1: Computational mesh for the flow around the unit sphere decomposed into 1024 subdomains.
• Figure 2: Vortex structures behind a unit sphere at Re 300 at times 2.5, 25, 45, 50, 60, 65, 71, 75, 80, 100, 125, 150, 175, 200 s.
• Figure 3: Cumulative number of PCG iterations over all time steps for Re = 100 (transient solution, left) and Re = 300 (periodic solution, right). Iterations terminated using ${\|{\bf r}_{k}\|}/{\|{\bf r}_{0}\|}$ (denoted by '$\|{\bf r}_{0}\|$') and ${\|{\bf r}_{k}\|}/{\|{\bf b}\|}$ ('$\|{\bf b}\|$') stopping criteria, initial guess taken as a zero vector ('${\bf 0}$') and as the approximate solution from the previous time step ('${\bf x}^\mathrm{prev}$').
• Figure 4: Norm of the initial and final residual over all time steps for Re = 100 (transient solution, left) and Re = 300 (periodic solution, right). Iterations terminated using ${\|{\bf r}_{k}\|}/{\|{\bf r}_{0}\|}$ (denoted as '$\|{\bf r}_{0}\|$') and ${\|{\bf r}_{k}\|}/{\|{\bf b}\|}$ ('$\|{\bf b}\|$') stopping criteria. The time from which the statistics are computed is marked by a vertical line for Re = 300.
• Figure 5: Drag and lift coefficients over all time steps for Re = 100 (transient solution, left) and Re = 300 (periodic solution, right). The lines for different stopping criteria visually overlap, with relative difference in most of the iterations below $10^{-6}$.