Multistep and multistage boundary integral methods for the wave equation

We describe how time-discretized wave equation in a homogeneous medium can be solved by boundary integral methods. The time discretization can be a multistep, Runge-Kutta, or a more general multistep-multistage method. The resulting convolutional system of boundary integral equations falls in the family of convolution quadratures of Ch. Lubich. In this work our aim is to discuss a new technique for efficiently solving the discrete convolutional system and to present large scale 3D numerical experiments with a wide range of time-discretizations that have up to now not appeared in print. One of the conclusions is that Runge-Kutta methods are often the method of choice even at low accuracy; yet, in connection with hyperbolic problems BDF (backward difference formulas) have been predominant in the literature on convolution quadrature.