Numerically derived boundary conditions on artificial boundaries

We consider partial differential equations in an infinite domain in which an artificial boundary B is introduced in order to restrict the computational domain to the region bounded by B. The nonlocal boundary condition on B is determined for equations of the type ▽2u + k2u = 0 in a separable coordinate system, and compared with two methods in which the boundary condition is approximated. One method uses the free space Greenʹs function directly and does not involve the evaluation of surface integrals. The other method, in which the boundary condition is derived from the solution of the Dirichlet problem in the domain exterior to B, is considered by several authors in the literature. Using Laplaceʹs equation in two dimensions, numerical results show that Greenʹs function approach is accurate, and that the boundary condition can be computed readily with standard numerical packages.