Revisiting the spectral analysis for high-order spectral discontinuous methods
The spectral analysis is a basic tool to characterise the behaviour of any convection scheme. By nature, the solution projected onto the Fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem. In this paper, we...
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| Veröffentlicht in: | Journal of computational physics Jg. 337; S. 379 - 402 |
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| Abstract | The spectral analysis is a basic tool to characterise the behaviour of any convection scheme. By nature, the solution projected onto the Fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem. In this paper, we wish to revisit such analysis, focusing attention on two key points. The first point concerns the effects of time integration on the spectral analysis. It is shown with standard high-order Finite Difference schemes dedicated to aeroacoustics that the time integration has an effect on the required number of points per wavelength. The situation depends on the choice of the coupled schemes (one for time integration, one for space derivative and one for the filter) and here, the compact scheme with its eighth-order filter seems to have a better spectral accuracy than the considered dispersion-relation preserving scheme with its associated filter, especially in terms of dissipation. Secondly, such a coupled space–time approach is applied to the new class of high-order spectral discontinuous approaches, focusing especially on the Spectral Difference method. A new way to address the specific spectral behaviour of the scheme is introduced first for wavenumbers in [0,π], following the Matrix Power method. For wavenumbers above π, an aliasing phenomenon always occurs but it is possible to understand and to control the aliasing of the signal. It is shown that aliasing depends on the polynomial degree and on the number of time steps. A new way to define dissipation and dispersion is introduced and applied to wavenumbers larger than π. Since the new criteria recover the previous results for wavenumbers below π, the new proposed approach is an extension of all the previous ones dealing with dissipation and dispersion errors. At last, since the standard Finite Difference schemes can serve as reference solution for their capability in aeroacoustics, it is shown that the Spectral Difference method is as accurate as (or even more accurate) than the considered Finite Difference schemes. |
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| AbstractList | The spectral analysis is a basic tool to characterise the behaviour of any convection scheme. By nature, the solution projected onto the Fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem. In this paper, we wish to revisit such analysis, focusing attention on two key points. The first point concerns the effects of time integration on the spectral analysis. It is shown with standard high-order Finite Difference schemes dedicated to aeroacoustics that the time integration has an effect on the required number of points per wavelength. The situation depends on the choice of the coupled schemes (one for time integration, one for space derivative and one for the filter) and here, the compact scheme with its eighth-order filter seems to have a better spectral accuracy than the considered dispersion relation preserving scheme with its associated filter, especially in terms of dissipation. Secondly, such a coupled space time approach is applied to the new class of high-order spectral discontinuous approaches, focusing especially on the Spectral Difference method. A new way to address the specific spectral behaviour of the scheme is introduced first for wavenumbers in [0,pi], following the Matrix Power method. For wavenumbers above pi, an aliasing phenomenon always occurs but it is possible to understand and to control the aliasing of the signal. It is shown that aliasing depends on the polynomial degree and on the number of time steps. A new way to define dissipation and dispersion is introduced and applied to wavenumbers larger than it. Since the new criteria recover the previous results for wavenumbers below it, the new proposed approach is an extension of all the previous ones dealing with dissipation and dispersion errors. At last, since the standard Finite Difference schemes can serve as reference solution for their capability in aeroacoustics, it is shown that the Spectral Difference method is as accurate as (or even more accurate) than the considered Finite Difference schemes. The spectral analysis is a basic tool to characterise the behaviour of any convection scheme. By nature, the solution projected onto the Fourier basis enables to estimate the dissipation and the dispersion associated with the spatial discretisation of the hyperbolic linear problem. In this paper, we wish to revisit such analysis, focusing attention on two key points. The first point concerns the effects of time integration on the spectral analysis. It is shown with standard high-order Finite Difference schemes dedicated to aeroacoustics that the time integration has an effect on the required number of points per wavelength. The situation depends on the choice of the coupled schemes (one for time integration, one for space derivative and one for the filter) and here, the compact scheme with its eighth-order filter seems to have a better spectral accuracy than the considered dispersion-relation preserving scheme with its associated filter, especially in terms of dissipation. Secondly, such a coupled space–time approach is applied to the new class of high-order spectral discontinuous approaches, focusing especially on the Spectral Difference method. A new way to address the specific spectral behaviour of the scheme is introduced first for wavenumbers in [0,π], following the Matrix Power method. For wavenumbers above π, an aliasing phenomenon always occurs but it is possible to understand and to control the aliasing of the signal. It is shown that aliasing depends on the polynomial degree and on the number of time steps. A new way to define dissipation and dispersion is introduced and applied to wavenumbers larger than π. Since the new criteria recover the previous results for wavenumbers below π, the new proposed approach is an extension of all the previous ones dealing with dissipation and dispersion errors. At last, since the standard Finite Difference schemes can serve as reference solution for their capability in aeroacoustics, it is shown that the Spectral Difference method is as accurate as (or even more accurate) than the considered Finite Difference schemes. |
| Author | Vasseur, Xavier Vanharen, Julien Sagaut, Pierre Boussuge, Jean-François Puigt, Guillaume |
| Author_xml | – sequence: 1 givenname: Julien orcidid: 0000-0002-0408-2467 surname: Vanharen fullname: Vanharen, Julien email: julien.vanharen@cerfacs.fr organization: Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique (CERFACS), 42 avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France – sequence: 2 givenname: Guillaume surname: Puigt fullname: Puigt, Guillaume email: guillaume.puigt@cerfacs.fr organization: Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique (CERFACS), 42 avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France – sequence: 3 givenname: Xavier surname: Vasseur fullname: Vasseur, Xavier email: xavier.vasseur@isae.fr organization: ISAE-SUPAERO, 10 avenue Edouard Belin, BP 54032, 31055 Toulouse Cedex 4, France – sequence: 4 givenname: Jean-François surname: Boussuge fullname: Boussuge, Jean-François email: jean-francois.boussuge@cerfacs.fr organization: Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique (CERFACS), 42 avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France – sequence: 5 givenname: Pierre surname: Sagaut fullname: Sagaut, Pierre email: pierre.sagaut@univ-amu.fr organization: Aix Marseille Univ, CNRS, Centrale Marseille, M2P2 UMR 7340, 13451 Marseille, France |
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| Cites_doi | 10.1016/j.jcp.2011.07.013 10.1016/j.jcp.2007.06.001 10.1016/0021-9991(92)90324-R 10.1007/s10915-011-9505-3 10.1016/j.compfluid.2013.12.007 10.1016/j.jcp.2003.09.003 10.1006/jcph.1999.6227 10.1016/j.jcp.2003.09.012 10.1016/j.jcp.2005.06.024 10.1007/s10915-010-9391-0 10.1016/j.cma.2011.08.019 10.1007/s10915-008-9201-0 10.1016/j.jcp.2006.01.024 10.1007/s10915-006-9113-9 10.1016/0021-9991(91)90264-L 10.1006/jcph.1993.1142 10.1006/jcph.2002.7041 10.1007/s10915-010-9420-z 10.1016/j.jcp.2005.08.033 10.1023/A:1025896119548 10.1016/j.jcp.2008.10.042 10.1006/jcph.1996.0091 10.1016/j.jcp.2009.07.036 10.1016/0021-9991(89)90183-6 10.1137/1.9781611970876 10.1016/j.jcp.2008.12.038 10.2514/2.1665 10.1142/S0218202503002568 10.1016/j.jcp.2010.03.027 10.1007/s10915-014-9882-5 10.1007/s10915-009-9339-4 10.1006/jcph.2002.7082 10.1051/meca/2013056 |
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| Keywords | Space–time spectral analysis Matrix Power Method Spectral discontinuous Aliasing Aeroacoustics Finite Difference Space-time spectral analysis |
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| SubjectTerms | Aeroacoustics Aliasing Computational physics Convection Dissipation Energy dissipation Engineering Sciences Finite Difference Finite difference method Finite element analysis Fluids mechanics Fourier transforms Linear equations Mathematical analysis Matrix Power Method Mechanics Space–time spectral analysis Spectra Spectral discontinuous Studies Time integration |
| Title | Revisiting the spectral analysis for high-order spectral discontinuous methods |
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