Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T06:07:48.239Z Has data issue: false hasContentIssue false
  • English
  • Français

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

Published online by Cambridge University Press:  20 August 2015

Xiaogang Deng ,
Meiliang Mao ,
Guohua Tu ,
Hanxin Zhang  and
Yifeng Zhang
Xiaogang Deng*
Affiliation:
State Key Laboratory of Aerodynamics, Aerodynamics Research & Development Center, Mianyang, 621000, P.R. China
Meiliang Mao*
Affiliation:
State Key Laboratory of Aerodynamics, Aerodynamics Research & Development Center, Mianyang, 621000, P.R. China
Guohua Tu*
Affiliation:
State Key Laboratory of Aerodynamics, Aerodynamics Research & Development Center, Mianyang, 621000, P.R. China
Hanxin Zhang*
Affiliation:
State Key Laboratory of Aerodynamics, Aerodynamics Research & Development Center, Mianyang, 621000, P.R. China
Yifeng Zhang*
Affiliation:
State Key Laboratory of Aerodynamics, Aerodynamics Research & Development Center, Mianyang, 621000, P.R. China
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this article is to summarize our recent progress in high-order and high accurate CFD methods for flow problems with complex grids as well as to discuss the engineering prospects in using these methods. Despite the rapid development of high-order algorithms in CFD, the applications of high-order and high accurate methods on complex configurations are still limited. One of the main reasons which hinder the widely applications of these methods is the complexity of grids. Many aspects which can be neglected for low-order schemes must be treated carefully for high-order ones when the configurations are complex. In order to implement high-order finite difference schemes on complex multi-block grids, the geometric conservation law and block-interface conditions are discussed. A conservative metric method is applied to calculate the grid derivatives, and a characteristic-based interface condition is employed to fulfil high-order multi-block computing. The fifth-order WCNS-E-5 proposed by Deng is applied to simulate flows with complex grids, including a double-delta wing, a transonic airplane configuration, and a hypersonic X-38 configuration. The results in this paper and the references show pleasant prospects in engineering-oriented applications of high-order schemes.

Keywords

76M20WCNScomplex configurationsgeometric conservation lawconservative metric methodscharacteristic-based interface conditions
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 4 , April 2012 , pp. 1081 - 1102
Copyright
Copyright © Global Science Press Limited 2012

References

[1]Wang, Z.J., High-Order Methods for the Euler and Navier-Stokes Equations on Unstructured Grids, Progress in Aerospace Sciences, 43 (2007), pp. 1–41.Google Scholar
[2]Shu, C.-W., High-Order Finite Difference and Finite Volume WENO Schemes and Discontinuous Gakerkin Methods for CFD, Int. J. Comput. Fluid Dyn., 17 (2003), pp. 107–118.Google Scholar
[3]Cheng, J., and Shu, C.-W., High Order Schemes for CFD: a Review, Chinese J. Comput. Phys., 26(5) (2009), pp. 633–655.Google Scholar
[4]Ekaterinaris, J.A., High-Order Accurate, Low Numerical Diffusion Methods for Aerodynamics, Progress in Aerospace Sciences, 41 (2005), pp. 192–300.Google Scholar
[5]Lele, S.K., Compact Finite Difference Schemes with Spectral-Like Resolution, J. Comput. Phys., 103 (1992), pp. 16–42.CrossRefGoogle Scholar
[6]Deng, X., Maekawa, H., and Shen, Q., A Class of High Order Dissipative Compact Schemes, AIAA Paper, 96–1972, (1996).Google Scholar
[7]Visbal, M.R., and Gaitonde, D.V., High-Order Accurate Methods for Complex Unsteady Subsonic Flows, AIAA Journal, 37(10) (1999), pp. 1231–1239.CrossRefGoogle Scholar
[8]Deng, X., Maekawa, H., Compact High-Order Accurate Nonlinear Schemes, J. Comput. Phys., 130 (1997), pp. 77–91.Google Scholar
[9]Deng, X., and Zhang, H., Developing High-Order Accurate Nonlinear Schemes, J. Comput. Phys., 165 (2000), pp. 22–44.CrossRefGoogle Scholar
[10]Deng, X., High-Order Accurate Dissipative Weighted Compact Nonlinear Schemes, Science in China (Serial A), 45(3) (2002), pp. 356–370.Google Scholar
[11]Liu, X., Deng, X., and Mao, M., High-Order Behaviors of Weighted Compact Fifth-Order Nonlinear Schemes, AIAA Journal, 45(8) (2007), pp. 2093–2097.Google Scholar
[12]Sumi, T., Kurotaki, T., and Hiyama, J., Generalized Characteristic Interface Conditions with High-Order Interpolation Method, AIAA Paper, 2008–752, (2008).Google Scholar
[13]Gaitonde, D.V., and Visbal, M.R., Padé-Type High-Order Boundary Filters for the Navier-Stokes Equations, AIAA Journal, 18(11) (2000), pp. 2103–2112.CrossRefGoogle Scholar
[14]Delf, J.W., Sound Generation From Gust-Airfoil Interaction Using CAA-Chimera Method, AIAA Paper, 2001–2136, (2001).Google Scholar
[15]Sherer, S.E., Gordnier, R.E., and Visbal, M.R., Computational Study of a UCAV Configuration Using a High-Order Overset-Grid Algorithm, AIAA Paper, 2008–626, (2008).Google Scholar
[16]Fujii, K., Nonomura, T., and Tsutsumi, S.Toward Accurate Simulation and Analysis of Strong Acoustic Wave Phenomena-A Rview Fom the Eperience of Our Study on Rocket Problems, Int. J. Numer. Meth. Fluids, 64 (2010), pp. 1412–1432.Google Scholar
[17]Rai, M.M., A Relaxation Approach to Patched Grid Calculations with the Euler Equations J. Comput. Phys., 66 (1986), pp. 99–131.Google Scholar
[18]Lerat, A., and Wu, Z.N., Stable Conservative Multidomain Treatments for Implicit Euler Equations, J. Comput. Phys., 123 (1996), pp. 45–64.CrossRefGoogle Scholar
[19]Huan, X., Hicken, J.E., and Zingg, D.W., Interface and Boundary Schemes for High-Order Methods, AIAA Paper 2009–3658, (2009).Google Scholar
[20]Hicken, J.E., and Zingg, D.W., Parallel Newton-Krylov Solver for the Euler Equations Dis-cretized Using Simultaneous-Approximation Terms, AIAA Journal, 46(11) (2008), pp. 2773–2786.Google Scholar
[21]Kreiss, H.-O., and Scherer, G., Finite Element and Finite Difference Methods for Hyperbolic Partial Differential Equations, Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, (1974).Google Scholar
[22]Mattsson, K., Boundary Procedures for Summation-by-Parts Operators, SIAM J. Sci. Comput., 18(1) (2003), pp. 133–153.Google Scholar
[23]Olsson, P., Summation by Parts, Projections, and Stability. I, Math. Comput., 64(211) (1995), pp. 1035–1065.CrossRefGoogle Scholar
[24]Carpenter, M.H., Gottlieb, D., and Abarbanel, S., Time-Stable Boundary Conditions for Finite-Difference Schemes Solving Hyperbolic Systems J. Comput. Phys., 111 (1994), pp. 220–236.Google Scholar
[25]Nordström, J. and Carpenter, M.H., Boundary and Interface Conditions for High-Order Finite-Difference Methods Applied to the Euler and Navier-Stokes Equations, J. Comput. Phys., 148 (1999), pp. 621–645.Google Scholar
[26]Carpenter, M.H., Nordstrom, J., and Gottlieb, D., A Stable and Conservative Interface Treatment of Arbitrary Spatial Accuracy, SIAM J. Sci. Comput., 148 (1999), pp. 341–365.Google Scholar
[27]Strand, B., Summation by Parts for Finite Difference Approximations for d/dx, J. Comput. Phys., 110 (1994), pp. 47–67.Google Scholar
[28]Jurgens, H.M. and Zingg, D.W., Numerical Solution of the Time-Domain Maxwell Equations Using High-Accuracy Finite-Difference Methods, SIAM J. Sci. Comput., 22(5) (2001), pp. 1675–1696.Google Scholar
[29]Kim, J.W., and Lee, D.J., Characteristic Interface Conditions for Multi-Block High-Order Computation on Singular Structured Grid, AIAA Paper, 2003–3122, (2003).Google Scholar
[30]Sumi, T., Kurotaki, T., and Hiyama, J., Generalized Characteristic Interface Conditions for Accurate Multi-Block Computation, AIAA Paper, 2006–1272, (2006).Google Scholar
[31]Sumi, T., Kurotaki, T., and Hiyama, J., Practical Multi-Block Computation with Generalized Characteristic Interface Conditions Around Complex Geometry, AIAA Paper, 2007–4471, (2007).Google Scholar
[32]Deng, X., Mao, M., and Tu, G., et al., Extending the fifth-order weighted compact nonlinear scheme to complex grids with characteristic-based interface conditions, AIAA Journal, 48(12) (2010), pp. 2840–2851.CrossRefGoogle Scholar
[33]Fu, D., Ma, Y., Analysis of Super Compact Finite Difference Method and Application to Simulation of Vortex-Shock Interaction, Int. J. Numer. Meth. Fluids, 36(7) (2001), pp. 773–805.Google Scholar
[34]Uzgoren, E., Sim, J., Shyy, W., Marker-Based, 3-D Adaptive Cartesian Grid Method for Multiphase Flow Around Irregular Geometries, Commun. Comput. Phys., 5 (2009), pp. 1–41.Google Scholar
[35]Yee, H.C., Explicit and Implicit Multidimensional Compact High-Resolution Shock-Capturing Methods: Formulation, J. Comput. Phys., 131 (1997), pp. 216–232.Google Scholar
[36]Sjogreen, B., Yee, H.C., Variable High Order Multiblock Overlapping Grid Methods for Mixed Steady and Unsteady Multiscale Viscous Flows, Commun. Comput. Phys., 5 (2009), pp. 730–744.Google Scholar
[37]Osher, S., Efficient Implementation of High Order Accurate Essentially Non-Oscillatory Shock Capturing Algorithms Applied to Compressible Flow, Numerical Methods for Compressible Flows-Finite Difference, Element and Volume Techniques. Anaheim, Ca, UAS: ASME, New York, NY, UAS, (1986), pp. 127–128.Google Scholar
[38]Chakravarthy, S.R., Harten, A., and Osher, S., Essentially Non-Oscillatory Shock-Capturing Schemes of Arbitrarily-High Accuracy, AIAA-86-0339 (1986).Google Scholar
[39]Shu, C.-W., High Oder Weighted Essentially Non-Oscillatory Schemes for Convection Dominated Problems, SIAM Rev., 51 (2009), pp. 82–126.Google Scholar
[40]Cockburn, B., and Shu, C.-W., The Runge-Kutta Discontinuous Galerkin Finite Element Method for Conservation Laws V: Multidimensional Systems, J. Comput. Phys., 141 (1998), pp. 199–224.Google Scholar
[41]Bassi, F., and Rebay, S., Numerical Evaluation of Two Discontinuous Galerkin Methods for the Compressible Navier-Stokes Equations, Int. J. Numer. Meth. Fluids, 40(1) (2002), pp. 197–207.Google Scholar
[42]Dolejsi, V., Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows, Commun. Comput. Phys., 4 (2008), pp. 231–274.Google Scholar
[43]Luo, H., Baum, J.D., and Lohner, R., A Discontinous Galerkin Method Based on a Taylor Basis for the Compressible Flows on Arbitrary Grids, J. Comput. Phys., 227 (2008), pp. 8875–8893.Google Scholar
[44]Dumbser, M., Balsara, D.S., Toro, E.F., and Munz, C.D., A Unified Framework for the Construction of One-Sep Finite Volume and Discontinuous Galerkin Schemes on Unstructured Meshes, J. Comput. Phys., 227 (2008), pp. 8209–8253.Google Scholar
[45]Ferrari, A., Dumbser, M., Toro, E.F., Armanini, A., A New Stable Version of the SPH Method in Lagrangian Coordinates, Commun. Comput. Phys., 4 (2008), pp. 378–404.Google Scholar
[46]Grosso, G., Antuono, M., Toro, E.F., The Riemann Problem for the Dispersive Nonlinear Shallow Water Equations, Commun. Comput. Phys., 7 (2010), pp. 64–102.Google Scholar
[47]Dumbser, M., Toro, E.F., On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws, Commun. Comput. Phys., 10 (2011), pp. 635–671.Google Scholar
[48]Wang, L., and Mavriplis, D.J., Implicit Solution of the Unsteady Euler Equations for HighOrder Accurate Discontinuous Galerkin Discretiztions J. Comput. Phys., 225 (2007), pp. 1994–2015.Google Scholar
[49]Nastase, C.R., and Mavriplis, D.J., High-Order Discontinuous Galerkin Methods Using a Hp-Multigrid Approach, J. Comput. Phys., 213 (2006), pp. 330–357.Google Scholar
[50]Abgrall, R., Shu, C.-W., Development of Residual Distribution Schemes for the Discontinuous Galerkin Method: The Scalar Case with Linear Elements, Commun. Comput. Phys., 5 (2009), pp. 376–390.Google Scholar
[51]Xia, Y., Xu, Y., Shu, C.-W., Application of the Local Discontinuous Galerkin Method for the Allen-Cahn/Cahn-Hilliard System, Commun. Comput. Phys., 5 (2009), 821–835.Google Scholar
[52]Zhang, Z.-T., Shu, C.-W., Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes, Commun. Comput. Phys., 5 (2009), pp. 836–848.Google Scholar
[53]Zhang, R., Zhang, M., Shu, C.-W., On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes, Commun. Comput. Phys., 9 (2011), pp. 807–827.Google Scholar
[54]Kroll, N., ADIGMA – A European Project on the Development of Adaptive Higher Order Variational Methods for Aerospace Applications, European Conference on Computational Fluid Dynamics, Eccomas CDF, (2006).Google Scholar
[55]Patera, A.T., A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion, J. Comput. Phys., 54 (1984), pp. 468.Google Scholar
[56]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., 215 (2006), pp. 454–472.Google Scholar
[57]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), pp. 41–58.Google Scholar
[58]Sun, Y., Wang, Z.J., and Liu, Y., High-Order Multi-Domain Spectral Difference Method for the Navier-Stokes Equations on Unstructured Hexahedral Grids, Commun. Comput. Phys., 2(2) (2007), pp. 310–333, and also AIAA Paper 2006-301, (2006).Google Scholar
[59]Kitamura, K., Shima, E., Fujimoto, K., Wang, Z.J., Performance of Low-Dissipation Euler Fluxes and Preconditioned LU-SGS at Low Speeds, Commun. Comput. Phys., 10 (2011), pp. 90–119.Google Scholar
[60]Kopriva, D.A., A Staggered-Grid Multidomian Spectral Method for the Compressible Navier-Stokes Equations, J. Comput. Phys., 143 (1998), pp. 125–158.Google Scholar
[61]Xu, K., Jin, C., A Unified Moving Grid Gas-Kinetic Method in Eulerian Space for Viscous Flow Computation, J. Comput. Phys., 222 (2007), pp. 155–175.Google Scholar
[62]Jin, C., Xu, K., Numerical Study of the Unsteady Aerodynamics of Freely Falling Plates, Commun. Comput. Phys., 3 (2008), pp. 834–851.Google Scholar
[63]Xu, K., A Gas-Kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method, J. Comput. Phys., 171 (2001) 289–335.Google Scholar
[64]Xu, K., Liu, H., A Multiple Temperature Kinetic Model and its Application to Near Continuum Flows, Commun. Comput. Phys., 4 (2008), pp. 1069–1085.Google Scholar
[65]Ni, G., Jiang, S., and Xu, K.Remapping-Free ALE-Type Kinetic Method for Flow Computations, J. Comput. Phys., 228 (2009), pp. 3154–3171.CrossRefGoogle Scholar
[66]Chen, Y., Jiang, S., An Optimization-Based Rezoning for ALE Methods, Commun. Comput. Phys., 4 (2008), pp. 1216–1244.Google Scholar
[67]Tam, C.K.W., and Webb, J.C., Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics, J. Comput. Phys., 107 (1993), pp. 262–281.Google Scholar
[68]Yee, H.C., Sandham, N. D., Hadjadj, A., Progress in the Development of a Class of Efficient Low Dissipative High Order Shock-Capturing Methods, Proceeding of the Symposium in Computational Fluid Dynamics for the 21st Century, July, (2000), Kyoto, Japan.Google Scholar
[69]Suresh, A., and Huynk, H.T., Accurate Monotonicity Preserving Scheme with Runge-Kutta Time Stepping, J. Comput. Phys., 136 (1997), pp. 83–99.Google Scholar
[70]Daru, V., and Tenaud, C., High Order One Step Monotonicity-Preserving Schemes for Unsteady Compressible Flow Calculations, J. Comput. Phys., 193, (2004), pp. 563–594.Google Scholar
[71]Trefethen, L.N., Group Velocity in Finite Difference Schemes, SIAM Rev., 24(2) (1982), pp. 113–136.Google Scholar
[72]Ma, Y., and Fu, D., Forth Order Accurate Compact Scheme with Group Velocity Control (GVC), Science in China (Serial A), 44(9) (2001), pp. 1197–1204.Google Scholar
[73]Ren, Y., Liu, M., and Zhang, H., A Characteristic-Wise Hybrid Compact-WENO Schemes for Solving Hyperbolic Conservations, J. Comput. Phys., 192 (2005), pp. 365–386.Google Scholar
[74]Adams, N.A., and Shariff, K., A High-Resolution Hybrid Compact-ENO Scheme for Shock-Turbulence Interaction Problems, J. Comput. Phys., 127 (1996), pp. 27.Google Scholar
[75]Pirozzoli, S., Conservative Hybrid Compact-WENO Schemes for Shock-Turbulence Interaction, J. Comput. Phys., 179 (2002), pp. 81–117.Google Scholar
[76]Wang, Z., and Huang, G., An Essentially Nonoscillatory High-Order Padé-Type (ENO-Padé) Scheme, J. Comput. Phys., 177 (2002), pp. 37–58.Google Scholar
[77]http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpwGoogle Scholar
[78]Visbal, R.M., Gaitonde, D.V., On the Use of Higher-Order Finite-Difference Schemes on Curvilinear and Deforming Meshes, J. Comput. Phys., 181 (2002), pp. 155–185.Google Scholar
[79]Nonomura, N., Iizuka, N., and Fujiji, K., Freestream and Vortex Preservation Properties of High-Order WENO and WCNS on Curvilinear Grids, Comput. Fluids, 39 (2010), pp. 197–214.Google Scholar
[80]Trulio, J.G., and Trigger, K.R., Numerical Solution of the One-Dimensional Hydrodynamic Equations in an Arbitrary Time-Dependent Coordinate System, Technical Report UCLR-6522, University of California Lawrence Radiation laboratory, (1961).Google Scholar
[81]Cheng, J., Shu, C.-W., A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations, Commun. Comput. Phys., 4 (2008), pp. 1008–1024.Google Scholar
[82]Qiu, J.-M., Shu, C.-W., Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation, Commun. Comput. Phys., 10 (2011), pp. 979–1000.Google Scholar
[83]Liu, W., Yuan, L., Shu, C.-W., A Conservative Modification to the Ghost Fluid Method for Compressible Multiphase Flows, Commun. Comput. Phys., 10 (2011), pp. 785–806.CrossRefGoogle Scholar
[84]Pulliam, T.H., and Steger, J.L., On Implicit Finite-Difference Simulations of Three-Dimensional Flow, AIAA Paper 78-10, (1978).Google Scholar
[85]Thomas, P.D., and Lombard, C.K., Geometric Conservation Law and Its Application to Flow Computations on Moving Grids, AIAA Journal, 17(10) (1979), pp. 1030–1037.Google Scholar
[86]Etienne, S., Garon, A., and Pelletier, D., Perspective on the Geometric Conservation Law and Finite Element Methods for ALE Simulations of Incompressible Flow, J. Comput. Phys., 228 (2009), pp. 2313–2333.Google Scholar
[87]Bassi, F., and Rebay, S., A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations, J. Sci. Comput., 131 (1997), pp. 267–279.Google Scholar
[88]Bassi, F., and Rebay, S., GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations, In Cockburn, Karniadakis and Shu, , eds., Discontinuous Galerkin Methods: Theory, Computation and Applications, pp. 197–208. Springer, Berlin, 2000.Google Scholar
[89]Thompson, K.W., Time Dependent Boundary Conditions for Hypersonic System, II, J. Comput. Phys., 89, (1990), pp. 439–461.Google Scholar
[90]Poinsot, T.J., and Lele, S.K., Boundary Conditions for Direction Simulations of Compressible Viscous Flows, J. Comput. Phys., 101 (1992), pp. 104–129Google Scholar
[91]Spalart, P.R., and Allmaras, S.R., A One-Equation Turbulence Model for Aerodynamic Flows, AIAA Paper 92–0439, (1992).Google Scholar
[92]Zhang, H., and Zhuang, F., NND Schemes and Their Applications to Numerical Simulation of Two- and Three-Dimensional Flows, Adv. Appl. Mech., 29 (1991), pp. 193–256Google Scholar
[93]Zhang, H., Reggio, M., Trépanier, J. Y., Camarero, R., Discrete Form of the GCL for Moving Meshes and its Implementation in CFD Shemes, Comput. Fluids, 22 (1993), pp. 9–23.Google Scholar
[94]Tu, G., Deng, X., Mao, M., A Staggered Non-Oscillatory Finite Difference Method for HighOrder Discretization of Viscous Terms, Acta Aerodynamica Sinica, 29(1) (2011), pp. 10–15.Google Scholar
[95]Wieting, A. R., Experimental Study of Shock Wave Interference Heating on a Cylindrical Leading Edge, NASA TM-100484, (1987).Google Scholar
[96]Fujii, K., Progress and Future Prospects of CFD in Aerospace-Wind Tunnel and Beyond, Progress in Aerospace Sciences, 41 (2005), pp. 455–470.Google Scholar
[97]Fujii, K.CFD Contributions to High-Speed Shock-Related Problems: Examples Today and New Features Tomorrow, Shock Waves, 18 (2008), pp. 145–154.Google Scholar
[98]Ishiko, K., Ohnishi, N., Ueno, K., et al., Implicit Large Eddy Simulation of Two-Dimensional Homogeneous Turbulence Using Weighted Compact Nonlinear Scheme, J. Fluid Eng., 131(6) (2009), pp. 1–14.Google Scholar
[99]Deng, X., Liu, X., Mao, M.et al., Advances in High-Order Accurate Weighted Compact Nonlinear Schemes. Adv. Mech., 37(3) (2007), pp.417–427.Google Scholar
[100]Deng, X., Mao, M., Tu, G., et al., Geometric Conservation Law and Applications to HighOrder Finite Difference Schemes with Stationary Grids, J. Comput. Phys., 230(4) (2011), pp. 1100–1115.Google Scholar
[101]Shen, Y., Zha, G., and Chen, X., High Order Conservative Differencing for Viscous Terms and the Application to Vortex-Induced Vibration Flows, J. Comput. Phys., 228 (2009), pp. 8283–8300.Google Scholar
[102]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), pp. 165–199.Google Scholar
[103]Kannan, R., and Wang, Z.J., LDG2: A Variant of the LDG Flux Formulation for the Spectral Volume Method, J. Sci. Comput., 46(2) (2010), pp. 314–328.Google Scholar
[104]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), pp. 2007–2021Google Scholar
[105]Sun, Y., Wang, Z.J., Liu, Y., Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids VI: Extension to Viscous Flow, J. Comput. Phys., 215 (2006), pp. 41–58.Google Scholar
[106]Zhang, Y., Deng, X., Mao, M.et al., Investigation of Convergence Acceleration for HighOrder Scheme (WCNS) in 2D Supersonic Flows, Acta Aerodynamica Sinica, 26(1) (2008), pp. 14–18.Google Scholar
[107]Parsani, M., Ghorbaniasl, G., Lacor, C., et al., An Implicit High-Order Spectral Difference Approach for Large Eddy Simulation, J. Comput. Phys., 229(14) (2010), pp. 5373–5393.Google Scholar
[108]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 Biomed. Eng., (Article online in advance of print) n/a. doi: 10.1002/cnm.1359.Google Scholar
[109]Sun, Y., Wang, Z.J., Liu, Y., Efficient Implicit Non-linear LU-SGS Approach for Compressible Flow Computation Using High-Order Spectral Difference Method, Commun. Comput. Phys., 5 (2009), pp. 760–778.Google Scholar
[110]Haga, T., Sawada, K., Wang, Z.J., An Implicit LU-SGS Scheme for the Spectral Volume Method on Unstructured Tetrahedral Grids, Commun. Comput. Phys., 6 (2009), pp. 978–996.Google Scholar
[111]Luo, H., Baum, J. D., Löhner|R., A P-Multigrid Discontinuous Galerkin Method for the Euler Equations on Unstructured Grids, J. Comput. Phys., 211 (2006), pp. 767–783.Google Scholar
[112]Fidkowski, K.J., Oliver, T.A., Lu, J., et al., P-Multigrid Solution of High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations, J. Comput. Phys., 207 (2005), pp. 92–113.Google Scholar
[113]Liang, C., Kannan, R., Wang, Z.J., A P-Multigrid Spectral Difference Method with Explicit and Implicit Smoothers on Unstructured Triangular Grids, Comput. Fluids, 38(2) (2009), pp. 254–265.Google Scholar
[114]Ronquist, E.M., Patera, A.T., Spectral Element Multigrid, I. Formulation and Numerical Results, J. Sci. Comput., 2(4) (1987), pp. 389–406.Google Scholar