Convolution Quadrature Galerkin Method for the Exterior Neumann Problem of the Wave Equation

The wave equation is important for many real-world applications in timedomain linear acoustics, including scattering from aircraft components and submarines and radiation from loudspeakers. The latter example forms the underlying motivation for the present study. Here the problem is to be solved on an unbounded exterior domain, and so the boundary integral method is a powerful tool for reducing this to an integral equation on the boundary of the radiating or scattering object. Time-domain boundary integral methods have been employed to solve wave propagation problems since the 1960s [Fr62]. Since then, increasing computer power has made numerical solutions possible over longer run times, and so long-time instabilities in the time marching numerical solutions have become evident [Bi99, Ry85]. A number of methods have been suggested to resolve this such as time averaging [Ry90] and modified time stepping [Bi99]. Using an implicit formulation with high order interpolation and quadrature was also found to give stable results for all practical purposes [Bl96, Do98]. Terrasse et al. [HaD03] obtained stable results using a Galerkin approach and used an energy identity to prove stability of the Galerkin approximation. A stable Burton–Miller type integral equation formulation has also been developed in the time domain [ChHa06, Er99]. In addition, the convolution quadrature method of Lubich [Lu88a, Lu88b] has been applied to a number of problems [Ab06, Ban08, Sc01] and has been shown to give stable numerical results. However, computations for the wave equation tend to be for either two-dimensional or very simple three-dimensional cases such as spheres.