Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T07:21:17.007Z Has data issue: false hasContentIssue false
  • English
  • Français

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms: Formulation and Analysis for Third Derivative Spatial Terms Using the LDG Discretization Procedure

Published online by Cambridge University Press:  20 August 2015

Ravi Kannan
Ravi Kannan*
Affiliation:
CFD Research Corporation, 215 Wynn Drive, Huntsville, AL 35805, USA
*
*Corresponding author.Email:[email protected]
Get access

Abstract

In this paper, we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume (SV) context; more specifically the emphasis is on handling equations containing third derivative terms. This formulation is based on the LDG (Local Discontinuous Galerkin) flux discretization method, originally employed for viscous equations containing second derivatives. A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation. The Fourier analysis was utilized for two purposes: firstly to eliminate all the unstable SV partitions, secondly to obtain the optimal SV partition. Numerical experiments are performed to illustrate the capability of this formulation. Since this formulation is extremely local, it can be easily parallelized and a h-p adaptation is relatively straightforward to implement. In general, the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries (KdV) type problems.

Keywords

65Spectral volumeLDGhigher spatial derivative termsKDVFourier analysis
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1257 - 1279
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114 (1994), 45–58.CrossRefGoogle Scholar
[2]Balakrishnan, K. and Menon, S., On the role of ambient reactive particles in the mixing and afterburn behind explosive blast waves, Combust. Sci. Technol., 182 (2010), 186–214.CrossRefGoogle Scholar
[3]Balakrishnan, K., Genin, F., Nance, D. V. and Menon, S., Numerical study of blast characteristics from detonation of homogeneous explosives, Shock Waves, 20(2) (2010), 147–162.Google Scholar
[4]Balakrishnan, K. and Menon, S., On turbulent chemical explosions into dilute aluminum particle clouds, Combust. Theor. Model., 14(4) (2010), 583–617.Google Scholar
[5]Barth, T. J. and Frederickson, P. O., High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA 90-0013, 1990.Google Scholar
[6]Cockburn, B. and Shu, C. W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16(3) (2001), 173–261.Google Scholar
[7]Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection diffusion system, SIAM J. Numer. Anal., 35 (1998), 2440–2463.CrossRefGoogle Scholar
[8]Delanaye, M. and Liu, Y., Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids, AIAA 99-3259-CP, 1999.Google Scholar
[9]Harris, R., Wang, Z. J. and Liu, Y., Efficient quadrature-free high-order spectral volume method on unstructured grids: theory and 2D implementation, J. Comput. Phys., 227 (2008), 1620–1642.Google Scholar
[10]Harris, R. and Wang, Z. J., High-order adaptive quadrature-free spectral volume method on unstructured grids, Comput. Fluids, 38 (2009), 2006–2025.CrossRefGoogle Scholar
[11]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes III, J. Comput. Phys., 71 (1987), 231–303.Google Scholar
[12]Kannan, R., An implicit LU-SGS spectral volume method for the moment models in device simulations: formulation in 1D and application to a p-multigrid algorithm, Int. J. Numer. Methods Biomedical Eng., accepted on 11 Oct. 2009, published online: 1 Feb. 2010.Google Scholar
[13]Kannan, R., An implicit LU-SGS s Volume method for the moment models in device simulations II: accuracy studies and performance enhancements using the penalty and the BR2 formulations, Int. J. Numer. Methods Biomedical Eng., accepted August 2010.Google Scholar
[14]Kannan, R. and Wang, Z. J., A study of viscous flux formulations for a p-multigrid spectral volume Navier stokes solver, J. Sci. Comput., 41(2) (2009), 165–199.Google Scholar
[15]Kannan, R. and Wang, Z. J., LDG2: a variant of the LDG flux formulation for the spectral volume method, J. Sci. Comput., published online 20th June 2010.Google Scholar
[16]Kannan, R. and Wang, Z. J., The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method, Comput. Fluids, article in press (doi:10.1016/j.compfluid.2010.07.006).CrossRefGoogle Scholar
[17]Kannan, R., High Order Spectral Volume and Spectral Difference Methods on Unstructured Grids, Ph.D Thesis, Iowa State University, 2008.Google Scholar
[18]Liang, C., Kannan, R. and Wang, Z. J., A-p-multigrid spectral difference method with explicit and implicit smoothers on unstructured grids, Comput. Fluids, 38(2) (2009), 254–265.CrossRefGoogle Scholar
[19] M.-S.Liou and Steffen, C., A new flux splitting scheme, J. Comput. Phys., 107 (1993), 23–39.Google Scholar
[20]Liu, Y., Vinokur, M. and Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems, J. Comput. Phys., 212 (2006), 454–472.Google Scholar
[21]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), 357–372.Google Scholar
[22]Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR 1 (1961), 267–279.Google Scholar
[23]Shu, C. W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1988), 1073–1084.CrossRefGoogle Scholar
[24]Sun, Y. and Wang, Z. J., Efficient implicit non-linear LU-SGS approach for compressible flow computation using high-order spectral difference method, Commun. Comput. Phys., 5 (2009), 760–778.Google Scholar
[25]Sun, Y., Wang, Z. J. and Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids VI: extension to viscous flow, J. Comput. Phys., 215 (2006), 41–58.Google Scholar
[26]Van, K. den Abeele, Broeckhoven, T. and Lacor, C., Dispersion and dissipation properties of the 1D spectral volume method and application to a p-multigrid algorithm, J. Comput. Phys., 224(2) (2007), 616–636.Google Scholar
[27]Van Leer, B., Towards the ultimate conservative difference scheme II: monotonicity and conservation combined in a second order scheme, J. Comput. Phys., 14 (1974), 361–376.Google Scholar
[28]Van Leer, B., Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’s method, J. Comput. Phys., 32 (1979), 101–136.Google Scholar
[29]Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys., 178 (2002), 210–251.Google Scholar
[30]Wang, Z. J. and Liu, Y., Spectral (finite) volume method for conservation lawson unstructured grids II: extension to two-dimensional scalar equation, J. Comput. Phys., 179 (2002), 665–697.Google Scholar
[31]Wang, Z. J. and Liu, Y., Spectral (finite) volume method for conservation lawson unstructured grids III: extension to one-dimensional systems, J. Sci. Comput., 20 (2004), 137–157.Google Scholar
[32]Wang, Z. J. and Liu, Y., Spectral (finite) volume method for conservation lawson unstructured grids IV: extension to two-dimensional Euler equations, J. Comput. Phys., 194 (2004), 716–741.Google Scholar
[33]Wang, Z. J. and Liu, Y., Extension of the spectral volume method to high-order boundary representation, J. Comput. Phys., 211 (2006), 154–178.Google Scholar
[34]Yan, J. and Shu, C. W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40(2) (2002), 769–791.Google Scholar
[35]Yan, J. and Liu, H., A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect, J. Comput. Phys., 215 (2006), 197–218.Google Scholar
[36]Zhang, M. and Shu, C. W., An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations, Math. Model. Methods Appl. Sci., 13 (2003), 395–413.Google Scholar