Concerning the cause of instability in time-stepping boundary element methods applied to the exterior acoustic problem

The boundary element method (BEM) in its simple form when solving the exterior acoustic problem in the frequency domain has difficulties at the frequencies of internal resonances of the closed structure. The corresponding time domain form of the exterior problem often exhibits instabilities in the time-stepping process. The link between these two features is investigated by relating the eigenvalues of the iterating matrix of the time-stepping process to damped frequencies. Numerical evidence from a problem with an analytic solution, and from a loudspeaker response are given. The suggested link implies that instability comes from numerical errors super-imposed on a more fundamental problem and may be best tackled through a time domain technique corresponding to the methods already available in the frequency domain.