Gray and Verosky have recently studied the reformulation of the N-state matrix representation of the time-dependent Schrödinger equation as an N-degrees of freedom classical Hamiltonian system. This opens the possibility of using in quantum dynamics numerical integrators originally devised for classical mechanics. When the Hamiltonian matrix is time-dependent, Gray and Verosky suggest the use of a Magnus approximation before reducing the quantum system to its classical format. We show that Magnus approximations are not necessary and suggest an alternative technique. With the new technique it is possible to obtain simple integrators of arbitrarily high orders of accuracy that can be applied to all matrix Schrödinger problems with a, possibly time-dependent, real Hamiltonian matrix. The connection between the new approach and high-order split-operator methods is studied.

Klein-Gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton (H-W) high-order absorbing boundary conditions (ABCs). Time integration is made by means of exponential splitting schemes that are efficient and easy to implement. In this way, it is possible to achieve a negligible error due to the time integration and to study the behavior of the absorption error. Numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided. The influence of the dispersion coefficient on the error is also studied.

Local absorbing boundary conditions with fifth order of absorption to approximate the solution of an initial value problem, for the spatially discretized wave equation, are considered. For the one dimensional case, it is proved necessary conditions for well posedness. Numerical experiments confirming well posedness and showing good results of absorption are included.

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