Runge--Kutta convolution coercivity and its use for time-dependent boundary integral equations
This work provides a theoretical foundation for using higher-order Runge-Kutta convolution quadrature in time-dependent boundary integral equations, addressing a known limitation of A-stable multistep methods.
The authors investigate which Runge-Kutta-based convolution quadrature methods preserve the coercivity property of temporal convolution operators, finding that the third-order Radau IIA method does so without restriction and that all algebraically stable Runge-Kutta methods do so with a Laplace domain shift. They apply this to analyze stability and convergence of time discretization for a nonlinear boundary integral equation from wave scattering.
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.