Optimality Theorems and Numerical Results on Block Preconditioners for Implicit Runge-Kutta Methods for PDEs in Engineering and Biophysics
In this work, the optimality properties of a broad class of block preconditioners for fully implicit Runge-Kutta methods are presented along with various numerical experiments that demonstrate the effectiveness of these preconditioners. For general linear hyperbolic and parabolic PDEs, order-optimality properties of these block pre- conditioners considered are proven for the timestep and spatial discretization size. Numerical analysis of the spectra and fields of values of the preconditioned systems is presented along with numerical experiments that show the performance of the pre- conditioners for improving GMRES convergence rates. Additionally, the preconditioners are shown to be order optimal with respect to timesteps and diffusion parameters for the stage 2 Radau IIA time-integrator ap- plied to stabilized advection-diffusion equations along with additional GMRES ex- periments coupled with numerical analytics of the appropriate spectra and fields of values. A new eigencluster-optimized block-diagonal preconditioner is proposed with a discussion behind the design and implementation of the preconditioners. Numeri- cal results are shown that demonstrate the strength of the optimized preconditioner, which frequently outperformed the established block preconditioners, including the gold standard preconditioner from the existing literature.