# Application of vector extrapolation methods to convergence to a steady state solution

I'm working on a fluid solver using dual-time stepping and everything works really well, except the convergence in pseudo-time is slow. I'd like to accelerate the convergence. I know multigrid methods are typically used, but I am thinking about exploring another approach that is, hopefully, easier to implement.

I started reading the literature on sequence convergence acceleration. My sequence in this case is the updated solution to the Navier-Stokes equations in pseudo-time and the sequence, if all goes well, converges to a constant field (alternatively, the pseudo-time derivative converges to zero) as an infinite number of steps are taken. My reading in this area has led me to vector extrapolation methods and the related algorithms, but I am unsure of a few points.

For sake of argument, consider the Reduced Rank Extrapolation (RRE) method. The algorithm can be found in Table 3.1 of this report. In summary, you collect a set of solution vectors (solve your equations over a time step and store the updated values). Then, compute the derivative of the solution (or really just store the increment from the fluid solver since that is what we compute anyway) and the resulting vectors become columns in a big matrix. Find the QR factorization of this big matrix and solve a least-squares problem, and use the result to extrapolate your solution vector. Hopefully your convergence accelerates.

First, can these actually help in this case? I think they can, but I'm a bit fuzzy on it all.

Second, can these methods be applied locally or do they need to be applied for all cells in the simulation simultaneously? Everything I have read just shows solution vectors. But I don't know if it's $nvar*ncells$ long, or if I can do the QR factorization in each cell to get an extrapolated vector. There's a big difference in cost between point-wise factorization and total-domain factorization in massively parallel simulations. The linked report says it can be done on subdomains, which seems to imply it is at least a "regional" (as in, not pointwise but at least within a processor) QR factorization.

• If it is not too late and if you still have difficulties, I believe I can help. (Vector extrapolation is a major research area of mine, and I have had experience also in applications to CFD problems.) Please let me know if I can be of help at this time. – user21683 Sep 14 '16 at 14:32
• @AvramSidi Amazingly timely because I'm about to try using this again for another application in CFD. I dropped it for this particular application but am about to pick it back up again for another. Any chance your user name is your actual name and I've been reading your papers for the past week? – tpg2114 Sep 14 '16 at 18:03