Accurate radiation boundary conditions for the two-dimensional wave equation on unbounded domains

A recursive sequence of radiation boundary conditions first given by Hagstrom and Hariharan [Appl. Numer. Math. 27 (1998) 403] for the time-dependent wave equation in a two-dimensional exterior region are re-derived based on direct application of the hierarchy of local boundary operators of Bayliss and Turkel [Commun. Pure Appl. Math. 33 (1980) 707] and a recursion relation for the expansion coefficients appearing in an asymptotic wave expansion. By introducing a decomposition into tangential Fourier modes on a circle we reformulate the sequence of local boundary conditions in integro-differential form involving systems of first-order temporal equations for auxiliary functions associated with each mode and the Fourier transform of the solution evaluated on the boundary. The auxiliary functions are recognized as residuals of the local boundary operators acting on the asymptotic wave expansion. Direct finite element implementations for the original local sequence of boundary conditions are compared to implementations of the Fourier transformed auxiliary functions. We show that both implementations easily fit into a standard finite element discretization provided that independent time integration algorithms are used for the interior and boundary equations with coupling through the boundary force vectors at each time step. For both of our direct and modal finite element implementations, the amount of work and storage is less than that required for the finite element calculation in the interior region within the boundary. One advantage of the tangential modal implementation is that far-field solutions may be computed separately for each Fourier mode without saving lengthy time-history data at interior points. Numerical studies confirm the progressive improvement in accuracy with increasing number of auxiliary functions included.