Recycling Krylov subspaces for CFD applications

The most popular iterative linear solvers in Computational Fluid Dynamics (CFD) calculations are restarted GMRES and BiCGStab. At the beginning of most incompressible flow calculations, the computation time and the number of iterations to converge for the pressure Poisson equation are quite high, since the initial guess is far from the solution. In this case, the BiCGStab algorithm, with relatively cheap but non-optimal iterations, may fail to converge for stiff problems. Thus, a more robust algorithm like GMRES, which guarantees monotonic convergence, is preferred. To reduce the large storage requirements of GMRES, a restarted version GMRES(m) or its variants is used in CFD applications. However, GMRES(m) can suffer from stagnation or very slow convergence. For this reason, we use the rGCROT method. rGCROT is an algorithm that improves restarted GMRES by recycling a selected subspace of the search space from one restart of GMRES(m) to the next as well as building and recycling this outer vector space from one problem to the next (subsequent time steps in this context). Such recycling of the search space generally reduces the run time of the iterative linear solvers for CFD problems where the matrix remains constant or changes slowly for subsequent systems. In the current work, we apply both GMRES and Bi-Lanczos based recycling to CFD simulations. For a turbulent channel flow problem with rGCROT we get comparable performance to BiCGStab. The rGCROT performance is still not as cheap as the BiCGStab algorithm in terms of both storage requirements and time to solution. Thus we have used a novel hybrid approach in which we get the benefits of the robustness of the GMRES(m) algorithm and the economical iterations of BiCGStab. For the first few time steps, the algorithm builds an outer vector space using rGCROT and then it recycles that space for the rBiCGStab solver. Time to solution with this approach is comparable to that for the BiCGStab algorithm with savings in the number of iterations and improved convergence properties. For a flow through porous media simulation, we get infinite performance gains over BiCGStab, since BiCGStab does not converge to a relative tolerance of 10−10 for the L norm of the residual. Overall, the hybrid approach also performs better than optimized rGCROT and GMRES(m).