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High Order Stable Finite Difference Methods for the Schrödinger Equation

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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.

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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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Correspondence to A. Nissen.


Appendix A: Proof of Lemma 2.1

We want to prove that \(w^{T} R w \geq\alpha h w_{1}^{2} \ \forall\) w T=(w 1,w 2,…) so that RαhE 0≥0. We have

where R 0 is a positive definite 3×3 matrix


We will need that \(\tilde{R}_{0} = R_{0} - E_{2} - \alpha E_{0} \geq0\), for some α>0, where

\(E_{0} = \operatorname{diag}(1,0,\ldots)\) and α should be determined. Furthermore, we write A 1=A 2E 0, A 1≥0. Then we have

Now let w T=(x T,y T), where x has only three components while y T=(y 1,y 2,…). Then

In order to determine α, we compute the eigenvalues λ i , i=1,2,3, of \(\tilde{R}_{0}\), where λ 1=0, \(\lambda _{2,3} = -\frac{\alpha-\frac{24}{9}}{2} \pm\sqrt{ (\frac{\alpha -\frac{24}{9}}{2} )^{2} + \frac{4}{9} ( 5\alpha- \frac{18}{9} ) }\). We see from the expression under the square root that \(\tilde{R}_{0}\) is positive semi-definite for \(\alpha\leq\frac{18}{45} = 0.4\). Thus, we have proved Lemma 2.1 with α=0.4.

Appendix B: C(0) for Dirichlet Boundary Conditions, Order 6

The matrix C(0) for Dirichlet boundary conditions, order 6, is given by

and is non-singular.

Appendix C: SBP Operators, Order 2

Below are the SBP operators, Q, P, A, and S for the 2nd order discretization, where Q=P −1(−AE 0 S).

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Nissen, A., Kreiss, G. & Gerritsen, M. High Order Stable Finite Difference Methods for the Schrödinger Equation. J Sci Comput 55, 173–199 (2013).

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