Asymptotic and Exact Radiation Boundary Conditions for Time-Dependent Scattering

Asymptotic and exact local radiation boundary conditions first derived by Hagstrom and Hariharan are reformulated as an auxiliary Cauchy problem for linear first-order systems of ordinary equations on the boundary for each harmonic on a circle or sphere in two- or three-dimensions, respectively. With this reformulation, the resulting radiation boundary condition involves first-order derivatives only and can be computed efficiently and concurrently with standard semi-discrete finite element methods for the near-field solution without changing the banded/sparse structure of the finite element equations. In 3D, with the number of equations in the Cauchy problem equal to the mode number, this reformulation is exact. If fewer equations are used, then the boundary conditions form uniform asymptotic approximations to the exact condition. Furthermore, using this approach, we formulate accurate radiation boundary conditions for the two-dimensional unbounded problem on a circle. Numerical studies of time-dependent radiation and scattering are performed to assess the accuracy and convergence properties of the boundary conditions when implemented in the finite element method. The results demonstrate that the new formulation has dramatically improved accuracy and efficiency for time domain simulations compared to standard boundary treatments.