Implementation and Evaluation of Multigrid Linear Solvers into Extended Magnetohydrodynamic Codes for Petascale Computing

Extended magnetohydrodynamic (MHD) codes are used to model the large, slow-growing instabilities that are projected to limit the performance of the International Thermonuclear Experimental Reactor (ITER). The multiscale nature of the extended MHD equations requires an implicit approach. However, the current linear solvers needed for the implicit algorithm scale poorly, because the resultant matrices are so ill-conditioned. The most successful scalable parallel processor solvers to date are multigrid solvers; the application of multigrid techniques to a set of equations whose fundamental modes are dispersive waves represents a promising approach. This project will implement the multigrid preconditioners and the Generalized Minimal Residual (GMRES) solver from Lawrence Livermore National Laboratory into extended MHD codes. Phase I will apply existing solver packages to the positive definite matrices in NIMROD, one of the extended MHD codes. Phase II will perform similar testing processes with M3D, another extended MHD code that uses a potential variable formalism on an unstructured grid. Then, the multi-level solvers will be optimized specifically for the extended MHD non-symmetric matrices on high-order finite element grids. Commercial Applications and other Benefits as described by the awardee: A successful development should allow for the efficient usage of future petascale computers at the National Leadership Facilities: Oak Ridge National Laboratory, Argonne National Laboratory, and the National Energy Research Scientific Computing Center.