Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schrödinger Equation
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 to include stability and accuracy at nonconforming 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.
KeywordsSchrödinger equation Finite difference methods Summation-by-parts operator Stability Grid interface
- 2.Stout, Q., De Zeeuw, D., Gombosi, T., Groth, C., Marshall, H., Powell, K.: Adaptive blocks: A high-performance data structure. In: Proceedings of the 1997 ACM/IEEE SC97 Conference (1997) Google Scholar
- 12.Reula, O.: Numerical treatment of interfaces in quantum dynamics (2011). arXiv:1103.5448v1 [quant-ph]
- 17.Nissen, A., Kreiss, G., Gerritsen, M.: High order stable finite difference methods for the Schrödinger equation. Technical report nr 2011-014, Department of Information Technology, Uppsala University (2011) Google Scholar