Abstract:In this talk we discuss the use of Runge-Kutta convolution quadratures of convolutions arising from time-domain boundary integral formulations of linear hyperbolic problems. We present some new results on the convergence analysis of such methods and also present numerous numerical examples.
It is well-known that order reduction can occur when discretizing stiff differential systems by Runge-Kutta methods. This phenomenon for parabolic systems and corresponding sectorial convolution quadratures has been analysed by Lubich & Ostermann in 1993. We extend these results to the hyperbolic case, that is to operators whose Laplace transform is bounded polynomially only in the right half complex plane.
Recently a need to refine the analysis for operators that are differently bounded in sectors of the right half-plane than in the whole half-plane, has been recognised in relation to applications coming from acoustics and electromagnetism. We present a different type of analysis for this class of problems and obtain a refined result that has some surprising consequences.
Numerical experiments and applications coming from solving time-domain boundary integral equations of acoustic and electromagnetic scattering confirm the theoretical results.
This is joint work with Christian Lubich (University of Tuebingen), Jens Markus Melenk (TU Vienna), Stefan Sauter, Alexander Veit (University of Zurich), and Jonas Ballani (MPI Leipzig).