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

TL;DR

This paper presents an approach to significantly accelerate the solution of a sequence of large linear systems in unsteady flow simulations by combining adaptive BDDC, Krylov subspace recycling, and deflation techniques, achieving over 40% time savings.

Contribution

It introduces an integrated method combining adaptive BDDC, Krylov subspace recycling, and deflation to efficiently solve sequences of linear systems in unsteady flow simulations, improving computational performance.

Findings

Achieved over 40% reduction in computational time.

Demonstrated effectiveness on flow around a unit sphere.

Validated approach with extensive parallel supercomputer experiments.

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.