Discretizing the wave equation in time by a multistep or Runge-Kutta method results in a discrete convolution equation. A simple trick transforms the discrete convolution into a decoupled system of Helmholtz equations which can be solved in parallel. In the case of A-stable multistep methods, the corresponding wave numbers are strictly in the upper half of the complex plane. The wavenumbers in the case of A-stable Runge-Kutta methods are small matrices with eigenvalues in the upper half plane. The transformation between the discrete convolution and the decoupled system can be performed efficiently by a scaled FFT.
In the case of acoustic scattering in a homogeneous medium, the Helmholtz equations can be solved by a boundary integral method. We show that the resulting method is a perturbation of convolution quadrature introduced by Ch. Lubich when applied to a retarded potential representation of the solution of the wave equation. Using the convolution quadrature theory, we give complete stability and convergence results for the multistep case and numerical results for both the Runge-Kutta and multistep approaches.
This is joint work with Stefan Sauter.