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On the recent advances of spectral analysis for systems arising from fully-implicit RK methods

Michal Outrata

TL;DR

The paper addresses spectral analysis of matrices arising from fully implicit Runge-Kutta methods applied to linear time-dependent PDEs. It presents two parallel analytic frameworks—a polynomial approach based on $F_{\tau M^{-1}K}(\lambda)$ and a matrix approach based on $X_{\mu_k}$—and shows their equivalence using Kronecker-product structure and the pencil $\tau K-\mu M$. The eigenvalues of the preconditioned system $\mathcal{P}^{-1}\mathcal{A}$ organize into $s$ branches parameterized by $\mu_k$, with connections between the polynomial and matrix formulations. The work highlights implications for GMRES convergence bounds and outlines future work combining BLT theory and Schwarz-Christoffel mapping to generalize estimates to broader spatial operators.

Abstract

This work deals with two groups of spectral analysis results for matrices arising in fully implicit Runge-Kutta methods used for linear time-dependent partial differential equations. These were applied for different formulations of the same problem and used different tools to arrive at results that do not immediately coincide. We show the equivalence of the results as well as the equivalence of the approaches, unifying the two directions.

On the recent advances of spectral analysis for systems arising from fully-implicit RK methods

TL;DR

The paper addresses spectral analysis of matrices arising from fully implicit Runge-Kutta methods applied to linear time-dependent PDEs. It presents two parallel analytic frameworks—a polynomial approach based on and a matrix approach based on —and shows their equivalence using Kronecker-product structure and the pencil . The eigenvalues of the preconditioned system organize into branches parameterized by , with connections between the polynomial and matrix formulations. The work highlights implications for GMRES convergence bounds and outlines future work combining BLT theory and Schwarz-Christoffel mapping to generalize estimates to broader spatial operators.

Abstract

This work deals with two groups of spectral analysis results for matrices arising in fully implicit Runge-Kutta methods used for linear time-dependent partial differential equations. These were applied for different formulations of the same problem and used different tools to arrive at results that do not immediately coincide. We show the equivalence of the results as well as the equivalence of the approaches, unifying the two directions.
Paper Structure (7 sections, 31 equations, 1 figure, 1 table)

This paper contains 7 sections, 31 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Formulas for the eigenproperties of the preconditioned system
  3. Polynomial approach
  4. Matrix approach
  5. Connecting the two
  6. Conclusion and future work
  7. Acknowledgment

Figures (1)

  • Figure 1: The spectra of the preconditioned system $\mathcal{P}^{-1}\mathcal{A}$ for the preconditioner $\mathcal{P}$ given in \ref{['eqn_secPolAprch_W2W1stencil_definition']} below, using the RadauIIA IRK method. The spatial operator is the Laplacian on an irregular domain $\Omega$ with various boundary conditions (Dirichlet, Neumann and Robin), discretized using conforming P1 FEM, see outrata2023irksstage for detailed description.

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Remark 3