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Spectral performance of RKDG methods

by V. Wheatley and R. Jeltsch and H. Kumar

(Report number 2011-01)

Abstract
The spectral properties of RKDG schemes are investigated by computing their approximate modified wavenumber behavior and by comparing numerically obtained spectra to that of an exact solution. The modified wavenumber behavior of high-order unlimited RKDG schemes is found to be excellent. In particular, the dispersive performance of the fourth-order scheme is remarkably good, with very little deviation from spectral behavior over the complete range of numerically resolved wavenumbers. The dissipation of this scheme is also very low, even at high wavenumbers. This behavior is confirmed by spectra from smooth numerical solutions. When limiting is required, however, the spectral performance of RKDG schemes tends to that of the first-order method at high wavenumbers. Thus in the vicinity of discontinuities, high-order RKDG methods exhibit high numerical dissipation due to the use of a limiter that reduces the polynomial order of the approximate solution to at most one.

Keywords:

@Techreport{WJK11_147,
  author = {V. Wheatley and R. Jeltsch and H. Kumar},
  title = {Spectral performance of RKDG methods},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-01.pdf },
  year = {2011}
}

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