A time-domain boundary element method for the 3D dissipative wave equation: Case of Neumann problems

The present article proposes a time-domain boundary element method (TDBEM) for the three-dimensional (3D) dissipative wave equation (DWE). Although the fundamental ingredients such as the Green's function for the 3D DWE have been known for a long time, the details of formulation and implementation for such a 3D TDBEM have been unreported yet to the author's best knowledge. The present formulation is performed truly in time domain on the basis of the time-dependent Green's function and results in a marching-on-in-time fashion. The main concern is in the evaluation of the boundary integrals. For this regard, weakly- and removable-singularities are carefully treated. The proposed TDBEM is checked through the numerical examples whose solutions can be obtained semi-analytically by means of the inverse Laplace transform. The results of the present TDBEM are satisfactory to validate its formulation and implementation for Neumann problems. On the other hand, the present formulation based on the ordinary boundary integral equation (BIE) is unstable for Dirichlet problems. The numerical analyses for the non-dissipative case imply that the instability issue can be partially resolved by using the Burton-Miller BIE even in the dissipative case.