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An Explicit Hermite-Taylor Method for the Schrödinger Equation

  • Daniel Appelö (a1), Gunilla Kreiss (a2) and Siyang Wang (a2)
Abstract
Abstract

An explicit spectrally accurate order-adaptive Hermite-Taylor method for the Schrödinger equation is developed. Numerical experiments illustrating the properties of the method are presented. The method, which is able to use very coarse grids while still retaining high accuracy, compares favorably to an existing exponential integrator – high order summation-by-parts finite difference method.

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*Corresponding author. Email addresses: appelo@math.unm.edu (D. Appelö), gunilla.kreiss@it.uu.se (G. Kreiss), siyang.wang@it.uu.se (S.Wang)
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Communicated by Chi-Wang Shu

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[1] Appelö D. and Hagstrom T.. On advection by Hermite methods. Paciffic Journal Of Applied Mathematics, 4(2):125139, 2011.
[2] Carpenter Mark H., Gottlieb David, and Abarbanel Saul. Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. J. Comput. Phys., 111(2):220236, 1994.
[3] Chen R. and Hagstrom T.. p-adaptive Hermite methods for initial value problems. ESAIM: Mathematical Modelling and Numerical Analysis, 46:545557, 2012.
[4] Chen X., Appelö D., and Hagstrom T.. A hybrid Hermite–discontinuous Galerkin method for hyperbolic systems with application to Maxwell's equations. Journal of Computational Physics, 257, Part A:501520, 2014.
[5] Goodrich J., Hagstrom T., and Lorenz J.. Hermite methods for hyperbolic initial-boundary value problems. Math. Comp., 75:595630, 2006.
[6] Hagstrom T. and Appelö D.. Solving PDEs with Hermite Interpolation. In Springer Lecture Notes in Computational Science and Engineering, 2015.
[7] Hochbruck M., Lubich C., and Selhofer H.. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19:15521574, 1998.
[8] Hochbruck M. and Ostermann A.. Exponential integrators. Acta Numerica, 19:209286, 2010.
[9] Kormann K. and Kronbichler M.. Parallel finite element operator application: graph partitioning and coloring. 2011 Seventh IEEE International Conference on eScience, pages 332339, 2011.
[10] Kormann K. and Larsson E.. A Galerkin radial basis function method for the Schrödinger equation. SIAM J. Sci. Comput., 35:A2832A2855, 2013.
[11] Kreiss H.-O. and Scherer G.. Finite element and finite difference methods for hyperbolic partial differential equations. In Mathematical Aspects of Finite Element in Partial Differential Equations. Academic Press, Inc., 1974.
[12] Mattsson K. and Nordström J.. Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys., 199:503540, 2004.
[13] Nissen Anna, Kreiss Gunilla, and Gerritsen Margot. High order stable finite difference methods for the Schrödinger equation. J. Sci. Comput., 55:173199, 2013.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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