Runge–Kutta convolution quadrature for the Boundary Element Method

Abstract

Time domain boundary element formulations can be established either directly in time domain or via Laplace or Fourier domain. Somewhere in between are the convolution quadrature based boundary element formulations which utilize the Laplace domain fundamental solution but establish a time stepping procedure. Up to now in applications mostly backward differential formulas of second order are used as the underlying multistep method. However, in recent mathematical literature also Runge–Kutta methods have been applied. Here, the use of Runge–Kutta methods is explained in detail and some numerical studies are given. In these studies the backward difference based procedures are compared to Runge–Kutta methods for a non-smooth problem. An ℓ2 norm of the error is used as the basis of comparison, the convergence of which is investigated theoretically as well. The results confirm that the usage of the new techniques is preferable with regard to less numerical oscillations in the solution and better representation of wave fronts.