Runge–Kutta convolution quadrature for the Boundary Element Method

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 ℓ2ℓ2 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.

► Runge–Kutta methods are applied in the Convolution Quadrature Method (RK-CQM).
► L2 convergence is proven for a special function.
► Numerical studies show that the RK-CQM improves the behavior at jumps in the solution.
► The application in a boundary element formulation shows the improvement of RK-CQM for the approximation of wave fronts.
► The numerical results show the higher convergence order of RK-CQM compared to a BDF2-CQM.