Speaker
Description
When using implicit Runge-Kutta methods (IRK) for solving linear parabolic or hyperbolic PDEs, solving the system of stage equations M * k = bis usually the computational bottleneck as the dimension of this problem for an s-stage IRK method becomes O(sn) where the spatial discretization dimension n can be very large. Hence the solution process requires the use of iterative solvers, whose convergence can be less than satisfactory. Moreover, due to the structure of the stage equations, the matrix M does not directly inherit any of the preferable properties of the spatial operator, making GMRES the go-to solver. Therefore we need a preconditioner and in [Neytcheva & Axelsson, 2020] and also [Howle et al., 2021, 2022 & 2024] a family of block preconditioners was utilized and numerically tested with promising results. These preconditioners are all based on the underlying Kronecker product structure of M and the fact that all manipulations with the s-by-s Butcher matrix A are cheap. Recently, in [Gander \& Outrata, 2024] and also [Dravins et al., 2023 \& 2024], two seemingly different approaches for a spectral analysis of the preconditioned systems were proposed. In the first paper, the emphasis is on using these results for estimating the preconditioned GMRES convergence, while in the others, the authors derive appealing asymptotic results on the spectrum clustering.
In this talk we will show the equivalence of the two approaches. We will also extend the techniques used for estimating the GMRES convergence profile to the framework used by Dravins et al., thus covering a larger family of parabolic or hyperbolic PDEs, for which we can get an insight into the preconditioned GMRES behavior. The main tools used are the Kronecker product structure, conformal mapping theory and Schwarz-Chrstoffel maps. We will demonstrate our results on several numerical examples.
