In this paper we extend the results from our earlier work on stable boundary closures for the Schrödinger equation using the summation-by-parts-simultaneous approximation term (SBP-SAT) method [8] to include stability and accuracy at non-conforming grid interfaces. Stability at the grid interface is shown by the energy method, and the estimates are generalized to multiple dimensions. The accuracy of the grid interface coupling is investigated using normal mode analysis for operators of 2nd and 4th order formal interior accuracy. We show that full accuracy is retained for the 2nd and 4th order operators. The accuracy results are extended to 6th and 8th order operators by numerical simulations, in which case two orders of accuracy is gained with respect to the lower order approximation close to the interface.

We derive a perfectly matched layer (PML) for the Schrödinger equation using a modal ansatz. We derive approximate error formulas for the modeling error from the outer boundary of the PML and the error from the discretization in the layer and show how to choose layer parameters so that these errors are matched and optimal performance of the PML is obtained. Numerical computations in 1D and 2D demonstrate that the optimized PML works efficiently at a prescribed accuracy for the zero potential case, with a layer of width less than a third of the de Broglie wavelength corresponding to the dominating frequency.

In this paper we extend the Summation-by-parts-simultaneous approximation term (SBP-SAT) technique to the Schrödinger equation. Stability estimates are derived and the accuracy of numerical approximations of interior order 2m, m = 1, 2, 3, are analyzed in the case of Dirichlet boundary conditions. We show that a boundary closure of the numerical approximations of order m lead to global accuracy of order m + 2. The results are supported by numerical simulations.

The perfectly matched layer (PML) technique is applied to a reactive scattering problem for accurate domain truncation. A two-dimensional model for dissociative adsorbtion and associative desorption of H(2) from a flat surface is considered, using a finite difference spatial discretization and the Arnoldi method for time-propagation. We compare the performance of the PML to that of a monomial complex absorbing potential, a transmission-free complex absorbing potential, and to exterior complex scaling. In particular, the reflection properties due to the numerical treatment are investigated. We conclude that the PML is accurate and efficient, especially if high accuracy is of significance. Moreover, we demonstrate that the errors from the PML can be controlled at a desired accuracy, enabling efficient numerical simulations.