Conveners
High order, variable step approximation of convolution equations
- María López Fernández (University of Málaga)
Description
We study the Generalized Convolution Quadrature (gCQ) based on Runge-Kutta methods to approximate the solution of an important class of convolution equations. The gCQ generalizes Lubich's original Convolution Quadrature to variable steps. High order versions of the gCQ have been developed in the last decade in a rather general setting, which includes applications to hyperbolic problems, where the convolution kernels are typically non smooth. However, in the case of convolutions with smoother, sectorial kernels, recent developments show that the available stability and convergence theory was suboptimal, both in terms of convergence order and regularity requirements of the data. Optimal results for the gCQ of the first order are now available, for a special important class of sectorial problems. We address here the generalization of this theory to high order and prove the same order of convergence as for the original Runge-Kutta based CQ with fixed steps, under the same regularity hypotheses about the data, and for arbitrary time meshes. Moreover, for data with known singularities of algebraic type, we show how to choose optimally graded time meshes in order to achieve maximal order of convergence, overcoming the well-known order reduction of the original CQ in these situations. We also show that a fast and memory reduced implementation of the gCQ is possible for this class of problems and illustrate our theoretical results with several numerical experiments.
