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.
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The authors would like to thank Katharina Kormann for help with the time-stepping method. This work has been partially financed by Anna Maria Lundins stipendiefond and the Swedish research council. The computations were performed on resources provided by SNIC through Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX) under Project p2005005.
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Nissen, A., Kreiss, G. & Gerritsen, M. Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schrödinger Equation. J Sci Comput 53, 528–551 (2012). https://doi.org/10.1007/s10915-012-9586-7
- Schrödinger equation
- Finite difference methods
- Summation-by-parts operator
- Grid interface