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Improving the High Order Spectral Volume Formulation Using a Diffusion Regulator

Published online by Cambridge University Press:  20 August 2015

Ravi Kannan  and
Zhijian Wang
Ravi Kannan*
Affiliation:
CFD Research Corporation, 215 Wynn Drive, Huntsville, AL 35805, USA
Zhijian Wang*
Affiliation:
Iowa State University, Howe Hall, Ames, IA 50011, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

The concept of diffusion regulation (DR) was originally proposed by Jaisankar for traditional second order finite volume Euler solvers. This was used to decrease the inherent dissipation associated with using approximate Riemann solvers. In this paper, the above concept is extended to the high order spectral volume (SV) method. The DR formulation was used in conjunction with the Rusanov flux to handle the inviscid flux terms. Numerical experiments were conducted to compare and contrast the original and the DR formulations. These experiments demonstrated (i) retention of high order accuracy for the new formulation, (ii) higher fidelity of the DR formulation, when compared to the original scheme for all orders and (iii) straightforward extension to Navier Stokes equations, since the DR does not interfere with the discretization of the viscous fluxes. In general, the 2D numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.

Keywords

65Diffusion regulationspectral volumehigh-orderRusanov fluxNavier Stokes equations
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 247 - 260
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]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(2) (2010), 2010–186.Google Scholar
[2]Balakrishnan, K. and Menon, S., On turbulent chemical explosions into dilute aluminium particle clouds, Combust. Theor. Model., 14(4) (2010), 2010–583.Google Scholar
[3]Balakrishnan, K., Nance, D. V. and Menon, S., Simulation of impulse effects from explosive charges containing metal particles, Shock Waves, 20(3) (2010), 2010–217.CrossRefGoogle Scholar
[4]Bassi, F. and Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier Stokes equations, J. Comput. Phys., 131 (1997), 267–279.CrossRefGoogle Scholar
[5]Brezzi, F., Manzini, G., Marini, D., Pietra, P. and Russo, A., Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differ. Equations, 16 (2000), 365–378.3.0.CO;2-Y>CrossRefGoogle Scholar
[6]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.Google Scholar
[7]Jaisankar, S. and Rao, S. V. Raghurama, Diffusion regulation for Euler solvers, J. Comput. Phys., 221 (2007), 577–599.Google Scholar
[8]Kannan, R., High order spectral volume method for moment models in semiconductor device simulations: formulation in 1D and application to a p-multigrid method, Int. J. Numer. Methods Biom. Eng., 27(9) (2011), 2011–1362.Google Scholar
[9]Kannan, R., High order spectral volume method for moment models in semiconductor device simulations II: Accuracy studies and performance enhancements using the penalty and the BR2 formulations, Int. J. Numer. Methods Biom. Eng., 27(5) (2011), 650–665.Google Scholar
[10]Kannan, R., An implicit LU-SGS spectral volume method for the moment models in device simulations III: accuracy enhancement using the LDG2 flux formulation for non-uniform grids, Int. J. Numer. Model. El., accepted.Google Scholar
[11]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.CrossRefGoogle Scholar
[12]Kannan, R. and Wang, Z. J., LDG2: a variant of the LDG viscous flux formulation for the spectral volume method, J. Sci. Comput., 46(2) (2011), 2011–314.CrossRefGoogle Scholar
[13]Kannan, R. and Wang, Z. J., The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method, Comput. Fluids, 39(10) (2010), 2010–2007.Google Scholar
[14]Kannan, R. and Wang, Z. J., Curvature and entropy based wall boundary condition for the high order spectral volume Euler solver, Comput. Fluids, 44(1) (2011), 2011–79.CrossRefGoogle Scholar
[15]Kannan, R., A high order spectral volume method for elastohydrodynamic lubrication problems: formulation and application of an implicit p-multigrid algorithm for line contact problems, Comput. Fluids, 48(1) (2011), 2011–44.Google Scholar
[16]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), 2009–254.Google Scholar
[17]Liou, M.-S. and Steffen, C., A new flux splitting scheme, J. Comput. Phys., 107 (1993), 23–39.CrossRefGoogle Scholar
[18]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
[19]Luo, H., Baum, J. D. and Löhner, R., A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids, J. Comput. Phys., 211 (2006), 767–783.Google Scholar
[20]Nastase, C. R. and Mavriplis, D. J., High-order discontinuous Galerkin methods using an hp-multigrid approach, J. Comput. Phys., 213 (2006), 330–357.Google Scholar
[21]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Com-put. Phys., 43 (1981), 357–372.Google Scholar
[22]Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Com-put. Math. Phys., USSR 1 (1961), 267–279.Google Scholar
[23]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., submitted.Google Scholar
[24]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
[25] C-Shu, W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1988), 1073–1084.Google Scholar
[26]Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys., 178 (2002), 210.Google Scholar
[27]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.Google Scholar
[28]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.Google Scholar
[29]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.Google Scholar
[30]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