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Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation

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Title: Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation
Authors: Harris, Rob1 rharris@iastate.edu, Wang, Z.J.1 zjw@iastate.edu, Liu, Yen2 liu@nas.nasa.gov
Source: Journal of Computational Physics. Jan2008, Vol. 227 Issue 3, p1620-1642. 23p.
Subjects: Spectral theory, Conservation laws (Mathematics), Gaussian quadrature formulas, Numerical integration
Abstract: Abstract: An efficient implementation of the high-order spectral volume (SV) method is presented for multi-dimensional conservation laws on unstructured grids. In the SV method, each simplex cell is called a spectral volume (SV), and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In the traditional implementation, Gauss quadrature formulas are used to approximate the flux integrals on all faces. In the new approach, a nodal set is selected and used to reconstruct a high-order polynomial approximation for the flux vector, and then the flux integrals on the internal faces are computed analytically, without the need for Gauss quadrature formulas. This gives a significant advantage over the traditional SV method in efficiency and ease of implementation. For SV interfaces, a quadrature-free approach is compared with the Gauss quadrature approach to further evaluate the accuracy and efficiency. A simplified treatment of curved boundaries is also presented that avoids the need to store a separate reconstruction for each boundary cell. Fundamental properties of the new SV implementation are studied and high-order accuracy is demonstrated for linear and non-linear advection equations, and the Euler equations. Several well known inviscid flow test cases are utilized to show the effectiveness of the simplified curved boundary representation. [Copyright &y& Elsevier]
Copyright of Journal of Computational Physics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 28010041
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  Data: Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation
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  Data: <searchLink fieldCode="AR" term="%22Harris%2C+Rob%22">Harris, Rob</searchLink><relatesTo>1</relatesTo><i> rharris@iastate.edu</i><br /><searchLink fieldCode="AR" term="%22Wang%2C+Z%2EJ%2E%22">Wang, Z.J.</searchLink><relatesTo>1</relatesTo><i> zjw@iastate.edu</i><br /><searchLink fieldCode="AR" term="%22Liu%2C+Yen%22">Liu, Yen</searchLink><relatesTo>2</relatesTo><i> liu@nas.nasa.gov</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Computational+Physics%22">Journal of Computational Physics</searchLink>. Jan2008, Vol. 227 Issue 3, p1620-1642. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Spectral+theory%22">Spectral theory</searchLink><br /><searchLink fieldCode="DE" term="%22Conservation+laws+%28Mathematics%29%22">Conservation laws (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Gaussian+quadrature+formulas%22">Gaussian quadrature formulas</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+integration%22">Numerical integration</searchLink>
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  Data: Abstract: An efficient implementation of the high-order spectral volume (SV) method is presented for multi-dimensional conservation laws on unstructured grids. In the SV method, each simplex cell is called a spectral volume (SV), and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In the traditional implementation, Gauss quadrature formulas are used to approximate the flux integrals on all faces. In the new approach, a nodal set is selected and used to reconstruct a high-order polynomial approximation for the flux vector, and then the flux integrals on the internal faces are computed analytically, without the need for Gauss quadrature formulas. This gives a significant advantage over the traditional SV method in efficiency and ease of implementation. For SV interfaces, a quadrature-free approach is compared with the Gauss quadrature approach to further evaluate the accuracy and efficiency. A simplified treatment of curved boundaries is also presented that avoids the need to store a separate reconstruction for each boundary cell. Fundamental properties of the new SV implementation are studied and high-order accuracy is demonstrated for linear and non-linear advection equations, and the Euler equations. Several well known inviscid flow test cases are utilized to show the effectiveness of the simplified curved boundary representation. [Copyright &y& Elsevier]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Journal of Computational Physics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.jcp.2007.09.012
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      – Code: eng
        Text: English
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        PageCount: 23
        StartPage: 1620
    Subjects:
      – SubjectFull: Spectral theory
        Type: general
      – SubjectFull: Conservation laws (Mathematics)
        Type: general
      – SubjectFull: Gaussian quadrature formulas
        Type: general
      – SubjectFull: Numerical integration
        Type: general
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      – TitleFull: Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation
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            NameFull: Harris, Rob
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              M: 01
              Text: Jan2008
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              Y: 2008
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