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A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II: Improving the Third Derivative Spatial Discretization Using the LDG2 Method

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:sunshekar@gmail.com
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Abstract

In this paper, the second in a series, we improve the discretization of the higher spatial derivative terms in a spectral volume (SV) context. The motivation for the above comes from [J. Sci. Comput., 46(2), 314-328], wherein the authors developed a variant of the LDG (Local Discontinuous Galerkin) flux discretization method. This variant (aptly named LDG2), not only displayed higher accuracy than the LDG approach, but also vastly reduced its unsymmetrical nature. In this paper, we adapt the LDG2 formulation for discretizing third derivative terms. A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and the LDG formulations. The results of the analysis showed that the LDG2 scheme (i) is stable for 2nd and 3rd orders and (ii) generates smaller dissipation and dispersion errors than the LDG formulation for all the orders. The 4th order LDG2 scheme is however mildly unstable: as the real component of the principal eigen value briefly becomes positive. In order to circumvent the above, a weighted average of the LDG and the LDG2 fluxes was used as the final numerical flux. Even a weight of 1.5% for the LDG (i.e., 98.5% for the LDG2) was sufficient to make the scheme stable. This weighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation. Numerical experiments are performed to validate the analysis. In general, the numerical results are very promising and indicate that the approach has a great potential for higher dimension Kortewegde Vries (KdV) type problems.

Keywords

65Spectral volumeLDG2LDGhigher spatial derivative termsKDVFourier analysis
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 3 , September 2012 , pp. 767 - 788
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Compute. Phys., 114 (1994), 4558.Google 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. Tech., 182 (2010), 186214.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), 147162.Google Scholar
[4]Balakrishnan, K. and Menon, S., On turbulent chemical explosions into dilute aluminum particle clouds, Combust. Theor. Model., 14(4 (2010), 583617.Google Scholar
[5]Barth, T. J. and Frederickson, P. O., High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper No. 90-0013, 1990.CrossRefGoogle Scholar
[6]Cockburn, B. and Shu, C. W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16(3 (2001), 173261.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), 24402463.Google Scholar
[8]Delanaye, M. and Liu, Y., Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids, AIAA Paper No. 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), 16201642.CrossRefGoogle Scholar
[10]Harris, R. and Wang, Z. J., High-order adaptive quadrature-free spectral volume method on unstructured grids, Comput. Fluids, 38 (2009), 20062025.Google 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.Google Scholar
[12]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), 13621375.Google Scholar
[13]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), 650665.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), 165199.Google Scholar
[15]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), 314328.Google Scholar
[16]Kannan, R. and Wang, Z. J., The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method, Compute. Fluids, 39(10 (2010), 20072021.Google Scholar
[17]Kannan, R., 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, Commun. Comput. Phys., 10(5 (2011), 12571279.CrossRefGoogle Scholar
[18]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), 7988.CrossRefGoogle Scholar
[19]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), 4453.Google Scholar
[20]Kannan, R. and Wang, Z. J., A high order spectral volume solution to the Burgers' equation using the Hopf-Cole transformation, Int. J. Numer. Methods Fluids, DOI: 10.1002/fld.2612.Google Scholar
[21]Kannan, R. and Wang, Z. J., Improving the high order spectral volume formulation using a diffusion regulator, Commun. Comput. Phys., 12 (2012), 247260.Google Scholar
[22]Kannan, R., High Order Spectral Volume and Spectral Difference Methods on Unstructured Grids, Thesis (Ph.D.)-Iowa State University, 2008, Publication Number: AAI3342262; ISBN: 9780549997177; Source: Dissertation Abstracts International, 70-01, 0437.Google Scholar
[23]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), 254265.Google Scholar
[24]Liou, M.-S. and Steffen, C., A new flux splitting scheme, J. Comput. Phys., 107 (1993), 2339.CrossRefGoogle Scholar
[25]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), 454472.CrossRefGoogle Scholar
[26]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), 357372.Google Scholar
[27]Rusanov, V. V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys., 1 (1961), 267279.Google Scholar
[28]Shu, C-W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1988), 10731084.Google Scholar
[29]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), 4158.Google Scholar
[30]Van den Abeele, K., 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), 616636.Google Scholar
[31]Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys., 178 (2002), 210.CrossRefGoogle Scholar
[32]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.CrossRefGoogle Scholar
[33]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.CrossRefGoogle Scholar
[34]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
[35]Wang, Z. J. and Liu, Y., Extension of the spectral volume method to high-order boundary representation, J. Comput. Phys., 211 (2006), 154178.Google Scholar
[36]Yan, J. and Shu, C.-W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40(2 (2002), 769791.Google Scholar
[37]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
[38]Zhang, M. and Shu, C. W., An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations, Math. Models Methods Appl. Sci., 13 (2003), 395-413.Google Scholar