Generalized convolution quadrature based on Runge-Kutta methods

In this paper, we develop the extit{Runge-Kutta generalized convolution quadrature} (gCQ) with variable time stepping for the numerical solution of convolution equations for time and space-time problems and present the corresponding stability and convergence analysis. For this purpose, some new theoretical tools such as extit{tensorial divided differences}, summation by parts with extit{Runge-Kutta differences} and a calculus for Runge-Kutta discretizations of generalized convolution operators such as an associativity property will be developed in this paper. Numerical examples will illustrate the stable and efficient behavior of the resulting discretization.