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Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Schrödinger Equations

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Abstract

In this paper, we study the superconvergence properties of the LDG method for the one-dimensional linear Schrödinger equation. We build a special interpolation function by constructing a correction function, and prove the numerical solution is superclose to the interpolation function in the \(L^{2}\) norm. The order of superconvergence is \(2k+1\), when the polynomials of degree at most k are used. Even though the linear Schrödinger equation involves only second order spatial derivative, it is actually a wave equation because of the coefficient i. It is not coercive and there is no control on the derivative for later time based on the initial condition of the solution itself, as for the parabolic case. In our analysis, the special correction functions and special initial conditions are required, which are the main differences from the linear parabolic equations. We also rigorously prove a \((2k+1)\)-th order superconvergence rate for the domain, cell averages, and the numerical fluxes at the nodes in the maximal and average norm. Furthermore, we prove the function value and the derivative approximation are superconvergent with a rate of \((k+2)\)-th order at the Radau points. All theoretical findings are confirmed by numerical experiments.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Lingling Zhou & Yan Xu

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

  4. School of Data and Computer Science, Sun Yat-Sen University, Guangzhou, 510006, Guangdong, People’s Republic of China

    Waixiang Cao

Authors
  1. Lingling Zhou
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  2. Yan Xu
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Correspondence to Yan Xu.

Additional information

Dedicated to Professor Chi-Wang Shu on the occasion of his 60th birthday.

Research of Yan Xu was supported by NSFC Grant Nos. 11371342, 11526212. Research of Zhimin Zhang was supported in part by the NSF Grant DMS-1419040 and NSFC Grants Nos. 91430216 and 11471031.

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Zhou, L., Xu, Y., Zhang, Z. et al. Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Schrödinger Equations. J Sci Comput 73, 1290–1315 (2017). https://doi.org/10.1007/s10915-017-0362-6

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  • DOI: https://doi.org/10.1007/s10915-017-0362-6

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