Generalized convolution quadrature based on Runge-Kutta methods

Abstract

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.

Generalised Convolution Quadrature with Runge‐Kutta methods for Acoustic Boundary Elements

“…The spatial discretisation is made as usual on a triangulation of the domain by introducing standard shape functions. As proposed in [3], the temporal discretisation is done with the generalised CQ. The algorithm is recalled here for the single layer potential (2).…”

“…The notationV denotes the Laplace transform of the single layer potential. The determination of the integration weights ω and the integration points s are given in [3]. The algorithm for (3) is analogously.…”

“…The generalisation to a non-uniform time mesh has been done by Lopez-Fernandez and Sauter [2]. This generalised CQ is used here with a Runge-Kutta method [3] for the single and double layer approach.…”

Convolution quadrature for the wave equation with impedance boundary conditions

“…However, if the right-hand side is not uniformly smooth and/or contains non-uniformly distributed variations in time, and/or consists of localized pulses, the use of adaptive time stepping becomes very important in order to keep the number of time steps reasonably small. The generalized convolution quadrature (gCQ) has been introduced in [24][25][26] and allows for variable time stepping.…”

“…In this paper, the generalized convolution quadrature (gCQ) is considered for the discretization of the RPIE. This method has been introduced in [24,26] for the implicit Euler time method and for the Runge-Kutta method in [25]. In contrast to the original CQ method the gCQ method allows for variable time stepping.…”