Preconditioned Multigroup GMRES Algorithms for the Variational Nodal Method

Combinations of three approaches are examined as options to replace the algorithms presently employed in the variational nodal code VARIANT. They are preconditioned Generalized Minimal Residual (GMRES) algorithms, parallelism in energy, and Wielandt acceleration. Together with partitioned matrix and Gauss-Seidel (GS) preconditioners, two GMRES algorithms are formulated to replace the upscattering iteration and facilitate energy parallelism and Wielandt acceleration. The GMRES algorithms are tested on two-dimensional thermal and fast reactor diffusion problems. The two GMRES algorithms yield higher efficiencies in energy group parallelization and Wielandt acceleration than simple parallelization of the existing GS algorithm. With preconditioning the GMRES algorithms reduce the total computing time by a factor of 2 to 4 and in some cases by a factor of >10. A multilevel iteration optimization scheme is investigated that automatically adjusts the relative error tolerance of the inner iterations according to the estimated convergence rate of the corresponding outer iterations and updates the Wielandt shift magnitude as the calculations progress. Numerical results based on large two-dimensional thermal and fast reactor diffusion problems demonstrate that automated optimization of the multilevel iterative processes reduces iteration numbers by as much as an order of magnitude.