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An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids

Published online by Cambridge University Press:  19 December 2014

Laiping Zhang ,
Ming Li ,
Wei Liu  and
Xin He
Laiping Zhang*
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang, Sichuan, 621000, China China Aerodynamics Research and Development Center, Mianyang, Sichuan, 621000, China
Ming Li
Affiliation:
China Aerodynamics Research and Development Center, Mianyang, Sichuan, 621000, China
Wei Liu
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang, Sichuan, 621000, China China Aerodynamics Research and Development Center, Mianyang, Sichuan, 621000, China
Xin He
Affiliation:
State Key Laboratory of Aerodynamics, Mianyang, Sichuan, 621000, China
*
*Email addresses: zhanglp_cardc@126.com (L. Zhang), mli@sina.com (M. Li), lw4992@gmail.com (W. Liu), fantasy2003@gmail.com (X. He)
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Abstract

A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit method was developed to solve two-dimensional Euler and Navier-Stokes equations by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed to solve the nonlinear system, while the linear system was solved with LU-SGS iteration. The effect of several parameters in the implicit scheme, such as the CFL number, the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated to evaluate the performance of convergence history. Several typical test cases were simulated, and compared with the traditional explicit Runge-Kutta (RK) scheme. Firstly the Couette flow was tested to validate the order of accuracy of the present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel was simulated and the effect of parameters was investigated also. Finally, the implicit algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder and the viscous flow in a square cavity. The numerical results demonstrated that the present implicit scheme can accelerate the convergence history efficiently. Choosing proper parameters would improve the efficiency of the implicit scheme. Moreover, in the same framework, the DG/FV hybrid schemes are more efficient than the same order DG schemes.

Keywords

65C2065L6065M0865M6068U2076M1076M12Implicit algorithmdiscontinuous Galerkin methodDG/FV hybrid SchemeNewton iterationGauss-Seidel iteration
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 1 , January 2015 , pp. 287 - 316
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high-order essentially non-oscillatory schemes III, J. Comput. Phys., 1987, 71: 231–303.CrossRefGoogle Scholar
[2]Jiang, G. and Shu, C-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 1996, 126: 202–228.CrossRefGoogle Scholar
[3]ReedW, H., Hill, T.R., Triangularmeshmethods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[4]Cockburn, B., Shu, C-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comp., 1989, 52: 411–435.Google Scholar
[5]Cockburn, B., Lin, S.Y., Shu, C-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 1989, 84: 90113.Google Scholar
[6]Cockburn, B., Hou, S., Shu, C-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comp., 1990, 54: 545–581.Google Scholar
[7]Cockburn, B., Shu, C-W., Runge-Kutta discontinuous Galerkin methods for convectiondominated problems, J. Sci. Comput., 2001: 173–261.Google Scholar
[8]Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. Comput. Phys., 2002, 178: 210–251.CrossRefGoogle Scholar
[9]Wang, Z.J. and Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation, J. Comput. Phys., 2002, 179: 665–697.Google Scholar
[10]Wang, Z.J. and Liu, Y., Spectral (finite) volumemethod for conservation laws on unstructured grids III: one-dimensional systems and partition optimization, J. Sci. Comput., 2004, 20: 137–157.Google Scholar
[11]Wang, Z.J., Zhang, L.P., Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional Euler equations, J. Comput. Phys., 2004, 194: 716–741.Google Scholar
[12]Liu, Y., Vinokur, M., and Wang, Z.J., Discontinuous spectral difference method for conservation laws on unstructured grids, J. Comput. Phys., 2006, 216: 780–801.CrossRefGoogle Scholar
[13]Venkatakrishnan, V., Allmaras, S.R., Kamenetskii, D.S., Johnson, F.T., Higher order schemes for the compressible Navier-Stokes equations, AIAA paper 2003–3987, 2003.Google Scholar
[14]May, G. and Jameson, A., A spectral differencemethod for the Euler and Navier-Stokes equations, AIAA paper 2006–304, 2006.Google Scholar
[15]Ekaterinaris, J.A., High-order accurate, low numerical diffusion methods for aerodynamics, Prog. Aero. Sci., 2005, 41: 192–300.Google Scholar
[16]Wang, Z.J., High-order methods for the Euler and Navier-Stokes equations on unstructured grids, Prog. Aero. Sci., 2007, 43: 1–41.CrossRefGoogle Scholar
[17]Cockburn, B., Karniadakis, G.E., Shu, C-W., Discontinuous Galerkin methods, Berlin: Springer, 2000.Google Scholar
[18]Thareja, R.R., Stewart, J.R., A point implicit unstructured grid solver for the Euler and Navier-Stokes equations, Inter. J. Numer. Meth. Fluids, 1989, 9: 405–425.CrossRefGoogle Scholar
[19]Luo, H., Baum, J.D. and Löhner, R., High-Reynolds number viscous computations using an unstructured-grid method, J. Aircraft, 2005, 42: 483–492.CrossRefGoogle Scholar
[20]He, L.X., Zhang, L.P., Zhang, H.X., A finite element/finite volume mixed solver on hybrid grids, Proceedings of the Fourth International Conference on Computational Fluid Dynamics, 10–14 July, 2006, Ghent, Belgium, edited by Deconinck, Herman and Dick, Erik, Springer Press, 2006: 695–700.Google Scholar
[21]Dumbser, M., Balsara, D.S., Toro, E.F., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructuredmeshes, J. Comput. Phys., 2008, 227: 8209–8253.Google Scholar
[22]Dumbser, M., Arbitrary high order PNPM schemes on unstructuredmeshes for the compressible Navier-Stokes equations, Computers and Fluids, 2010, 39: 60–76.Google Scholar
[23]Dumbser, M., Zanotti, O., Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, J. Comput. Phys., 2009, 228: 6991–7006.Google Scholar
[24]Luo, H., Luo, L., Norgaliev, R., Mousseau, V.A. and Dinh, N., A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, J. Comput. Phys., 2010, 229: 6961–6978.Google Scholar
[25]Luo, H., Luo, L.P., Ali, A., Norgaliev, R. and Cai, C., A parallel, reconstructed discontinuous Galerkin method for the compressible flows on arbitrary grids, Commun. Comput. Phy., 2011, 9(2): 363–389.Google Scholar
[26]Qiu, J.X., Shu, C-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case, J. Comput. Phys., 2003, 193: 115–135.Google Scholar
[27]Qiu, J.X., Shu, C-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two-dimensional case, Computers and Fluids, 2005, 34: 642–663.Google Scholar
[28]Zhang, L.P., Liu, W., He, L.X., Deng, X.G., Zhang, H.X., A class of DG/FV hybrid methods for conservation laws I: Basic formulation and one-dimensional systems, J. Comput. Phys., 2012, 231: 1081–1103.Google Scholar
[29]Zhang, L.P., Liu, W., He, L.X., Deng, X.G., Zhang, H.X., A class of DG/FV hybrid methods for conservation laws II: Two-dimensional Cases, J. Comput. Phys., 2012, 231: 1104–1120.Google Scholar
[30]Zhang, L.P., Liu, W., He, L.X., Deng, X.G., Zhang, H.X., A class of DG/FV hybrid methods for conservation laws III: Two-dimensional Euler equations, Commun. Comput. Phy., 2012, 12(1): 284–314.Google Scholar
[31]Huynh, H.T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA paper 2007–4079, 2007.Google Scholar
[32]Wang, Z.J., Gao, H.Y., A unifying lifting collocation penalty formulation for the Euler equations on mixed grids, AIAA paper 2009–0401, 2009.Google Scholar
[33]Luo, H., Baum, J.D., Löhner, R., A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 2008, 227: 8875–8893.Google Scholar
[34]Briley, W.R., McDonald, H., Solution of the three dimensional compressible Navier-Stokes equations by an implicit technique. In: Proceedings of the fourth international conference on numerical methods in fluid dynamics, Lecture Notes in Physics, vol.35. Berlin: Springer-Verlag; 1975.Google Scholar
[35]Beam, R.W., Warming, R.F., An implicit finite-difference algorithm for hyperbolic systems in conservation law form. J. Comput. Phys., 1976, 22: 87–109.Google Scholar
[36]Wang, L., Mavriplis, D.J., Implicit solution of the unsteady Euler equation for high-order accurate discontinuous Galerkin discretizations, J. Comput. Phys., 2007, 225:1994–2005.Google Scholar
[37]Mavriplis, D.J., Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes, J. Comput. Phys., 1998; 145:141–165.Google Scholar
[38]Bassi, F., Crivellini, A., Rebay, S., Savini, M., Discontinuous Galerkin solutions of the Reynoldsaveraged Navier-Stokes and kw turbulence model equations, Computers and Fluids, 2005, 34: 507–540.Google Scholar
[39]Saad, Y., Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 1986, 7: 865–884.CrossRefGoogle Scholar
[40]Rasetarinera, P., Hussaini, M.Y., An efficient implicit discontinuous Galerkinmethod, J. Comput. Phys., 2001, 172: 718–738.CrossRefGoogle Scholar
[41]Sharov, D., Nakahashi, K., Low speed preconditioning and LU-SGS scheme for 3D viscous flow computations on unstructured grids, AIAA paper 98–0614, 1998.Google Scholar
[42]Sun, Y.Z., Wang, Z.J., Liu, Y. and Chen, C.L., Efficient implicit LU-SGS algorithmfor high-order spectral differencemethod on unstructured hexahedral grids, AIAA paper 2007–0313, 2007.Google Scholar
[43]Yasue, K., Furudate, M., Ohnishi, N., Sawada, K., Implicit discontinuous Galerkin method for RANS simulation utilizing pointwise relaxation algorithm, Commun. Comput. Phys., 2010, 7(3): 510–533.Google Scholar
[44]Kanevsky, A., Carpenter, M.H., Gottlieb, D., Hesthaven, J.S., Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes, J. Comput. Phys., 2007, 225: 1753–1781.Google Scholar
[45]Ascher, U.M., Ruuth, S.J., Spiteri, R.J., Implicit-explicit Runge-Kutta methods for timedependent partial differential equations, Appl. Numer. Math., 1997, 25: 151–167.Google Scholar
[46]MacCormack, R.W., Implicit methods for fluid dynamics, Computers and Fluids, 2011, 41: 72–81.CrossRefGoogle Scholar
[47]Xia, Y.D., Luo, H., Nourgaliev, R., An implicit method for a reconstructed discontinuous Galerkin method on tetrahedron grids, AIAA paper 2012–834, 2012.Google Scholar
[48]Xia, Y.D., Luo, H., Frisbey, M., Nourgaliev, R., A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids, Computers & Fluids, 2014, 98: 134–151.Google Scholar
[49]Crivellini, A., Bassi, F., An implicit matrix-free discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations, Computers & Fluids, 2011, 50(1): 81–93.Google Scholar
[50]Nastase, C.R., Mavriplis, D.J., High-order discontinuous Galerkin methods using an hpmultigrid approach, J. Comput. Phys., 2006, 213: 330–357.Google Scholar
[51]Jameson, A., Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA paper 91–1596, 1991.Google Scholar
[52]Jameson, A. and Yoon, S., Lower-upper implicit schemes with multiple grids for the Euler equations, AIAA J., 1987. 25(7): 929–935.Google Scholar
[53]Wang, Z.J. and Gao, H.Y., A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys., 2009, 228: 8161–8186.Google Scholar
[54]Krivodonova, L., Berger, M., High-order accurate implementation of solid wall boundary condition in curved geometries, J. Comput. Phys., 2006, 211: 492–512.Google Scholar
[55]Ghia, U., Ghia, K.N., Shin, C.T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 1982, 48: 387–411.Google Scholar