Space–time spectral collocation method for Klein–Gordon equation

By using the Legendre–Laguerre collocation method, we can construct a spectral collocation scheme to solve the Klein–Gordon equation on the half-line. The Laguerre function collocation method (based on the Lagrange interpolation) in space and the Legendre–Gauss–Lobatto collocation method in time are used. A Newton iterative algorithm is provided. The numerical results demonstrate the high efficiency and accuracy of suggested algorithms.