Samenvatting
The spectral volume (SV) method was introduced in 2002 by Wang. Its further development, such as the extension to 2D and 3D and the application to the Euler and Navier-Stokes equations, was reported by Wang et al. , Liu et al. and Sun et al. . Recently, a more efficient, quadrature-free implementation of the SV method was presented in Harris et al. A number of high-order SV schemes for triangles and tetrahedrons were defined in Chen. The SV method is related to the discontinuous Galerkin method, see e.g. Cockburn and Shu, and the references therein, and to the spectral difference (SD) method, see e.g. Sun et al. All these methods use piecewise continuous polynomials to represent the solution. Moreover, in Van den Abeele et al. , it was shown that in 1D, the SV and SD methods are equivalent.\\
In Van den Abeele et al. and Van den Abeele and Lacor, the stability of the SV method was analyzed for 1D and 2D respectively. Several SV schemes that have been used in literature were found to suffer from weak instabilities, and stable SV schemes were proposed for 1D and 2D. In the present note, the stability of the SV method on 3D tetrahedral grids is analyzed using the matrix method, see e.g. Hirsch \cite{Hirsch88}. The second-order scheme and four families of third-order schemes are investigated. Although the second-order scheme is stable, a surprising result is that no stable scheme was found in any of the four families of third-order schemes.\\
In Van den Abeele et al. and Van den Abeele and Lacor, the stability of the SV method was analyzed for 1D and 2D respectively. Several SV schemes that have been used in literature were found to suffer from weak instabilities, and stable SV schemes were proposed for 1D and 2D. In the present note, the stability of the SV method on 3D tetrahedral grids is analyzed using the matrix method, see e.g. Hirsch \cite{Hirsch88}. The second-order scheme and four families of third-order schemes are investigated. Although the second-order scheme is stable, a surprising result is that no stable scheme was found in any of the four families of third-order schemes.\\
Originele taal-2 | English |
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Pagina's (van-tot) | 257-265 |
Aantal pagina's | 9 |
Tijdschrift | Journal of Computational Physics |
Volume | 228 |
Status | Published - 1 jan 2009 |