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A coercivity property of temporal convolution operators is an essential toolin the analysis of time-dependent boundary integral equations and their spaceand time discretisations. It is known that this coercivity property isinherited by convolution quadrature time discretisation based on A-stablemultistep methods, which are of order at most two. Here we study the questionas to which Runge--Kutta-based convolution quadrature methods inherit theconvolution coercivity property. It is shown that this holds without anyrestriction for the third-order Radau IIA method, and on permitting a shift inthe Laplace domain variable, this holds for all algebraically stableRunge--Kutta methods and hence for methods of arbitrary order. As anillustration, the discrete convolution coercivity is used to analyse thestability and convergence properties of the time discretisation of a non-linearboundary integral equation that originates from a non-linear scattering problemfor the linear wave equation. Numerical experiments illustrate the errorbehaviour of the Runge--Kutta convolution quadrature time discretisation.

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