A spectral element solution of the Klein–Gordon equation with high-order treatment of time and non-reflecting boundary

A spectral element (SE) implementation of the Givoli–Neta non-reflecting boundary condition (NRBC) is considered for the solution of the Klein–Gordon equation. The infinite domain is truncated via an artificial boundary BB, and a high-order NRBC is applied on BB. Numerical examples, in various configurations, concerning the propagation of a pressure pulse are used to demonstrate the performance of the SE implementation. Effects of time integration techniques and long term results are discussed. Specifically, we show that in order to achieve the full benefits of high-order accuracy requires balancing all errors involved; this includes the order of accuracy of the spatial discretization method, time-integrators, and boundary conditions.