# High Order Stable Finite Difference Methods for the Schrödinger Equation

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## Abstract

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 2*m*, *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.

### Keywords

Schrödinger equation High-order finite difference methods Summation-by-parts operators Numerical stability## Notes

### Acknowledgements

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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