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The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode

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Abstract

In this paper, we develop a multiscale local discontinuous Galerkin (LDG) method to simulate the one-dimensional stationary Schrödinger-Poisson problem. The stationary Schrödinger equation is discretized by the WKB local discontinuous Galerkin (WKB-LDG) method, and the Poisson potential equation is discretized by the minimal dissipation LDG (MD-LDG) method. The WKB-LDG method we propose provides a significant reduction of both the computational cost and memory in solving the Schrödinger equation. Compared with traditional continuous finite element Galerkin methodology, the WKB-LDG method has the advantages of the DG methods including their flexibility in h-p adaptivity and allowance of complete discontinuity at element interfaces. Although not addressed in this paper, a major advantage of the WKB-LDG method is its feasibility for two-dimensional devices.

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Author notes

  1. Wei Wang

    Present address: Center for Turbulence Research, Stanford University, Stanford, CA, 94305, USA

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Wei Wang & Chi-Wang Shu

Authors
  1. Wei Wang
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  2. Chi-Wang Shu
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Correspondence to Wei Wang.

Additional information

Research supported by NSF grants DMS-0510345 and DMS-0809086.

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Wang, W., Shu, CW. The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode. J Sci Comput 40, 360–374 (2009). https://doi.org/10.1007/s10915-008-9237-1

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  • DOI: https://doi.org/10.1007/s10915-008-9237-1

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