New applications of dynamic relaxation in advanced scientific computing

This work explores the implementation of highly efficient elliptic solvers in the Center for Extended MHD Modeling on vector supercomputer, the Cray XIE. Our simulation code spends the majority of its time inside these solvers, thus their efficiency is critical for good optimization. The solvers currently in use invoke Krylov methods and AMG preconditioning through the PETSc and Hypre libraries. These linear solvers have been designed for superscalar computers, and tend to perform poorly on vector architectures. In order to take advantage of the Cray XIE, one of the most powerful machines nowadays, the relaxation algorithms are revisited and studied. This type of algorithms, named Constant Relaxation since 5t and T are fixed for all timesteps, guarantees convergence without preconditioning and shows no stagnation, unlike GMRES. Although it takes more iterations to converge, the code eventually gets accelerated for each iteration is much cheaper and all operations are highly vectorizable. The code is even sped up by Dynamical Relaxation, introduced by varying St and r at each timestep. This idea is further extended to construct nonlinear solver for nonlinear PDEs by varying 5t and r in both temporal and spatial directions at each timestep. Therefore, this project has both allowed us to construct more efficient elliptic solvers on vector supercomputer; and led to new and interesting numerical results of mathematical interest.