An analysis of the finite-difference method for one-dimensional Klein–Gordon equation on unbounded domain

The numerical solution of the one-dimensional Klein–Gordon equation on an unbounded domain is analyzed in this paper. Two artificial boundary conditions are obtained to reduce the original problem to an initial boundary value problem on a bounded computational domain, which is discretized by an explicit difference scheme. The stability and convergence of the scheme are analyzed by the energy method. A fast algorithm is obtained to reduce the computational cost and a discrete artificial boundary condition (DABC) is derived by the Z-transform approach. Finally, we illustrate the efficiency of the proposed method by several numerical examples.