Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Runge-Kutta approximation for $C_0$-semigroups in the graph norm with applications to time domain boundary integral equations

Alexander Rieder, Francisco-Javier Sayas, Jens Markus Melenk

TL;DR

The paper develops a priori convergence theory for $A$-stable Runge-Kutta discretizations of abstract evolution equations with inhomogeneous boundary constraints, obtaining estimates in the graph norm of the generator in addition to the Banach-space norm. It extends existing results by introducing integrated and differentiated error estimates and by leveraging interpolation/Sobolev-tower machinery to bound boundary traces and time-integrals, thereby addressing $A_\star u$ in $[\mathcal X_0,\mathcal X_1]_{\mu}$ scales. The main contributions include Theorems 3.1–3.3, which yield integrated, differentiated, and strong-norm error bounds (including maximal dissipative settings), plus detailed lemmas on rational operator functions and discrete convolutions. Applications to time-domain boundary integral equations via convolution quadrature and to heat- and scattering-type problems illustrate the practical impact, with numerical experiments confirming predicted rates and highlighting end-time behavior and potential superconvergence in certain cases.

Abstract

We consider the approximation to an abstract evolution problem with inhomogeneous side constraint using $A$-stable Runge-Kutta methods. We derive a priori estimates in norms other than the underlying Banach space. Most notably, we derive estimates in the graph norm of the generator. These results are used to study convolution quadrature based discretizations of a wave scattering and a heat conduction problem.

Runge-Kutta approximation for $C_0$-semigroups in the graph norm with applications to time domain boundary integral equations

TL;DR

The paper develops a priori convergence theory for -stable Runge-Kutta discretizations of abstract evolution equations with inhomogeneous boundary constraints, obtaining estimates in the graph norm of the generator in addition to the Banach-space norm. It extends existing results by introducing integrated and differentiated error estimates and by leveraging interpolation/Sobolev-tower machinery to bound boundary traces and time-integrals, thereby addressing in scales. The main contributions include Theorems 3.1–3.3, which yield integrated, differentiated, and strong-norm error bounds (including maximal dissipative settings), plus detailed lemmas on rational operator functions and discrete convolutions. Applications to time-domain boundary integral equations via convolution quadrature and to heat- and scattering-type problems illustrate the practical impact, with numerical experiments confirming predicted rates and highlighting end-time behavior and potential superconvergence in certain cases.

Abstract

We consider the approximation to an abstract evolution problem with inhomogeneous side constraint using -stable Runge-Kutta methods. We derive a priori estimates in norms other than the underlying Banach space. Most notably, we derive estimates in the graph norm of the generator. These results are used to study convolution quadrature based discretizations of a wave scattering and a heat conduction problem.

Paper Structure

This paper contains 24 sections, 40 theorems, 222 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Operator equation, functional calculus, and Sobolev towers
  4. Runge-Kutta approximation and discrete stage derivative
  5. Error estimates
  6. The estimates of Alonso-Mallo and Palencia
  7. New results in this article
  8. Some computations related to the main theorems
  9. Some Lemmas regarding Runge-Kutta methods
  10. Estimates on rational functions of the operator
  11. Estimates on discrete convolutions
  12. Proofs
  13. Proof of Theorem \ref{['theorem:3.1']}
  14. Proof of Theorem \ref{['theorem:3.2']}
  15. Proof of Theorem \ref{['theorem:3.3']}
  16. ...and 9 more sections

Key Result

Proposition 2.1

If $A$ is the generator of a $C_0$-semigroup on a Banach space $\mathcal{X}$, then there exist constants $\omega \geq 0$ and $M\geq 1$ such that the spectrum $\sigma(A)$ of $A$ satisfies $\sigma(A) \subseteq \{z \in \mathbb{C}: \mathrm{Re}\,z \leq \omega\}$ and the resolvent satisfies the estimates

Figures (3)

  • Figure 8.1: Snapshots of the simulation at $t=2.85$, $t=4.45$, $t=5.0$, $t=5.6$, $t=12$
  • Figure 8.2: Performance of Radau IIA methods for the wave equation, cf. Section \ref{['sect:scattering_numerics']}
  • Figure 8.3: Convergence for the density $\widetilde{\lambda}$ for the heat conduction problem (cf. Section \ref{['sect:heat']}), comparing Radau IIA methods.

Theorems & Definitions (84)

  • Proposition 2.1
  • proof
  • Definition 2.2: Rational functions of operators
  • Definition 2.3: Sobolev towers
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 74 more