Transient Boundary Element Method and Numerical Evaluation of Retarded Potentials Ernst P. Stephan 1 , Matthias Maischak 2 , and Elke Ostermann 1 1 Leibniz Universit¨at Hannover, Institut f¨ ur Angewandte Mathematik, Am Welfengarten 1, 30167 Hannover, Germany {stephan,osterman}@ifam.uni-hannover.de 2 Brunel University, School of Information Systems, Computing & Mathematics John Crank Building, Uxbridge UB8 3PH, United Kingdom matthias.maischak@brunel.ac.uk Abstract. We discuss the modeling of transient wave propagation with the boundary element method (BEM) in three dimensions. The special structure of the fundamental solution of the wave equation leads to a close interaction of space and time variables in a so-called retarded time- argument. We give a detailed derivation of the discretization scheme and analyse a new kind of ”geometrical light cone” singularity of the re- tarded potential function. Moreover, we present numerical experiments that show these singularities. Keywords: retarded potential, transient boundary element method. 1 Introduction The simulation of sound radiation for automotive systems has become of great industrial interest and here the transient boundary element method is a powerful tool, especially for high frequency radiation phenomena. Nevertheless, the sim- ulation of three dimensional time dependent problems is a challenging task. Ha- Duong [1] was able to prove the unconditional stability of the transient boundary element method with Galerkin’s method in space and time. Hence, the question arises, why have so many instabilities in numerical experiments been reported. For a first kind integral equation with the retarded single layer potential we per- form a p-version boundary element method both in space and time. We address especially the question of how to gain an accurate quadrature method. Thus we analyse the potentials that arise in the computation of the Galerkin elements and show that a new kind of ”geometrical light cone” singularity can cause se- vere inaccuracies in the computation of the Galerkin stiffness matrix. Therefore, an accurate quadrature scheme which respects these singularities, e.g. by an appropriate grading, needs to be used for the computation of the matrix entries. The paper is structured as follows. First, we represent the solution u of the Dirichlet problem of the wave equation via a single layer ansatz with unknown density function p and derive a first kind integral equation for p on the surface Γ of the scatterer. For this integral equation we give a variational formulation which M. Bubak et al. (Eds.): ICCS 2008, Part II, LNCS 5102, pp. 321–330, 2008. c  Springer-Verlag Berlin Heidelberg 2008