Spectral Analysis of Implicit [math]-Stage Block Runge–Kutta Preconditioners

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A2047-A2072, June 2024.
Abstract. We analyze the recently introduced family of preconditioners in [M. M. Rana et al., SIAM J. Sci. Comput., 43 (2021), pp. S475–S495] for the stage equations of implicit Runge–Kutta methods for [math]-stage methods. We simplify the formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general [math]-stage method and use these to obtain convergence rate estimates for preconditioned GMRES for some common choices of the implicit Runge–Kutta methods. This analysis is based on understanding the inherent matrix structure of these problems and exploiting it to qualitatively predict and explain the main observed features of the GMRES convergence behavior, using tools from approximation and potential theory based on Schwarz–Christoffel maps for curves and close, connected domains in the complex plane. We illustrate our analysis with numerical experiments showing very close correspondence of the estimates and the observed behavior, suggesting the analysis reliably captures the essence of these preconditioners.