We consider the numerical solution of the linear Klein-Gordon equation in $\bbfR^2\times$ and $\bbfR^3\times$. An artificial boundary is introduced to obtain a bounded computational domain. On the given artificial boundary, the exact boundary condition and a series of approximating boundary conditions are constructed, which are called absorbing boundary conditions. By using either the exact or approximating boundary conditions on the artificial boundary, the original problem is reduced to either an equivalent or an approximately equivalent initial-boundary value problem on the bounded computational domain. The uniqueness of the approximate problem is then proved. The numerical results demonstrate that the method given in this paper is effective and feasible.
"Absorbing boundary conditions for the multidimensional Klein-Gordon equation." Commun. Math. Sci. 5 (3) 743 - 764, September 2007.