# Runge--Kutta convolution coercivity and its use for time-dependent boundary integral equations

## TL;DR

This paper investigates the inheritance of convolution coercivity by Runge--Kutta convolution quadrature methods, demonstrating that certain methods preserve this property and applying it to analyze stability and convergence in nonlinear boundary integral equations.

## Contribution

It proves that all algebraically stable Runge--Kutta methods, with a shift in the Laplace domain, inherit convolution coercivity, extending previous results limited to multistep methods.

## Key findings

- Radau IIA method of order three inherits coercivity without restrictions.
- All algebraically stable Runge--Kutta methods inherit coercivity with a shifted Laplace variable.
- Numerical experiments confirm the theoretical error and stability analysis.

## Abstract

A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretisations. It is known that this coercivity property is inherited by convolution quadrature time discretisation based on A-stable multistep methods, which are of order at most two. Here we study the question as to which Runge--Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge--Kutta methods and hence for methods of arbitrary order. As an illustration, the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretisation of a non-linear boundary integral equation that originates from a non-linear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge--Kutta convolution quadrature time discretisation.