Generalized convolution quadrature based on Runge-Kutta methods

Convolution equations for time and space-time problems have many important applications, e.g., for the modelling of wave or heat propagation via ordinary and partial differential equations as well as for the corresponding integral equation formulations. For their discretization, the convolution quadrature (CQ) has been developed since the late 1980’s and is now one of the most popular method in this field. However, the method and the theory are restricted to constant time stepping and only recently the implicit Euler generalized convolution quadrature (gCQ) has been developed which allows for variable time stepping. In this paper, we develop the gCQ for Runge-Kutta methods with variable time stepping and present the corresponding stability and convergence analysis. For this purpose, some new theoretical tools such as tensorial divided differences, summation by parts with 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.