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Dispersion and Dissipation Errors of Two Fully Discrete Discontinuous Galerkin Methods

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Abstract

The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K=kh→0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY, 12180, USA

    He Yang & Fengyan Li

  2. School of Mathematical Science, Xiamen University, Xiamen, Fujian, 361005, P.R. China

    Jianxian Qiu

Authors
  1. He Yang
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  2. Fengyan Li
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  3. Jianxian Qiu
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Correspondence to Fengyan Li.

Additional information

The research of F. Li and H. Yang is partially supported by NSF CAREER award DMS-0847241 and an Alfred P. Sloan Research Fellowship. The research of J. Qiu is partially supported by the National Science Foundation of China Grant No. 10931004 and ISTCP of China Grant No. 2010DFR00700.

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Yang, H., Li, F. & Qiu, J. Dispersion and Dissipation Errors of Two Fully Discrete Discontinuous Galerkin Methods. J Sci Comput 55, 552–574 (2013). https://doi.org/10.1007/s10915-012-9647-y

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