Runge--Kutta generalized Convolution Quadrature for sectorial problems

TL;DR

This paper advances the application of generalized convolution quadrature based on Runge--Kutta methods to sectorial problems, achieving optimal convergence rates and efficient implementation, especially for data with algebraic singularities.

Contribution

It proves that for certain sectorial problems, Runge--Kutta gCQ attains the same convergence order as classical CQ, even on general time meshes, and provides optimal mesh strategies for singular data.

Findings

Achieves optimal convergence order for sectorial problems.

Provides a fast, memory-efficient implementation of Runge--Kutta gCQ.

Demonstrates effectiveness through numerical experiments.

Abstract

We study the application of the generalized convolution quadrature (gCQ) based on Runge--Kutta methods to approximate the solution of an important class of sectorial problems. The gCQ generalizes Lubich's original convolution quadrature (CQ) to variable steps. High-order versions of the gCQ have been developed in the last decade, relying on certain Runge--Kutta methods. The Runge--Kutta based gCQ has been studied so far in a rather general setting, which includes applications to boundary integral formulations of wave problems. The available stability and convergence results for these new methods are suboptimal compared to those known for the uniform-step CQ, both in terms of convergence order and regularity requirements of the data. Here we focus on a special class of sectorial problems and prove that in these important applications it is possible to achieve the same order of convergence as for the original CQ, under the same regularity hypotheses on the data, and for very general time meshes. In the particular case of data with some known algebraic type of singularity, we also show how to choose an optimally graded time mesh to achieve convergence with maximal order, overcoming the well-known order reduction of the original CQ in these situations. An important advantage of the gCQ method is that it allows for a fast and memory-efficient implementation. We describe how the fast and oblivious Runge--Kutta based gCQ can be implemented and illustrate our theoretical results with several numerical experiments. The codes implementing the examples in this article are available in [13].