A $p$-version of convolution quadrature in wave propagation

We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the timestep size ($h$-method), we achieve accuracy by increasing the order of the method ($p$-method). We base this method on discontinuous Galerkin timestepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observes(root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the findings.