Runge–Kutta convolution quadrature for operators arising in wave propagation

Abstract

An error analysis of Runge–Kutta convolution quadrature is presented for a class of non-sectorial operators whose Laplace transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ ₀ and a polynomial bound ${\operatorname{O}(|s|^{\mu_1})}$ there, the stronger polynomial bound ${\operatorname{O}(s^{\mu_2})}$ in convex sectors of the form ${|\operatorname*{arg} s| \leq \pi/2-\theta}$ for θ > 0. The order of convergence of the Runge–Kutta convolution quadrature is determined by μ ₂ and the underlying Runge–Kutta method, but is independent of μ ₁. Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of Runge–Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical order of the Runge–Kutta method is attained away from the scattering boundary.