An Artificial Boundary Condition for the Numerical Computation of Scattering Waves

We consider the controllability method, which is proposed by Bristeau-GlowinskiPériaux [BGP98], for computing numerical solutions of the exterior problem for the Helmholtz equation. In the controllability method, we need to introduce an artificial boundary in order to reduce the computational domain to a bounded domain, and need to solve, in the bounded computational domain, the wave equation and an elliptic problem iteratively. We first introduce a new artificial boundary condition (ABC) for the wave equation, which is suitable for the controllability method. Our ABC is constructed by using the Dirichlet-to-Neumann (DtN) operator associated with the Helmholtz equation. We next discuss uniqueness for semi-discrete solution of the controllability method in the case when the artificial boundary is a circle. Then we need spectral properties of the DtN operator, which are deduced from some properties of the Hankel functions. We finally present numerical examples, which show that numerical solutions obtained by using our ABC are more accurate than those obtained by using another well-known ABC, and that by using our ABC, accurate numerical solutions are obtained whether the artificial boundary is large or small. These numerical results suggest that by using our ABC and by taking a small artificial boundary, we can reduce the computational costs. We consider the exterior problem for the Helmholtz equation: