Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver

摘要

We focus on robust and efficient iterative solvers for the pressure Poissonequation in incompressible Navier-Stokes problems. Preconditioned Krylovsubspace methods are popular for these problems, with BiCGStab and GMRES(m)most frequently used for nonsymmetric systems. BiCGStab is popular because ithas cheap iterations, but it may fail for stiff problems, especially early onas the initial guess is far from the solution. Restarted GMRES is better, morerobust, in this phase, but restarting may lead to very slow convergence.Therefore, we evaluate the rGCROT method for these systems. This methodrecycles a selected subspace of the search space (called recycle space) after arestart. This generally improves the convergence drastically compared withGMRES(m). Recycling subspaces is also advantageous for subsequent linearsystems, if the matrix changes slowly or is constant. However, rGCROTiterations are still expensive in memory and computation time compared withthose of BiCGStab. Hence, we propose a new, hybrid approach that combines thecheap iterations of BiCGStab with the robustness of rGCROT. For the first fewtime steps the algorithm uses rGCROT and builds an effective recycle space, andthen it recycles that space in the rBiCGStab solver. We evaluate rGCROT on aturbulent channel flow problem, and we evaluate both rGCROT and the new, hybridcombination of rGCROT and rBiCGStab on a porous medium flow problem. We seesubstantial performance gains on both problems.