Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T18:16:36.142Z Has data issue: false hasContentIssue false
  • English
  • Français

A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh

Published online by Cambridge University Press:  02 April 2015

Marc R. J. Charest ,
Clinton P. T. Groth  and
Pierre Q. Gauthier
Marc R. J. Charest*
Affiliation:
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada M3H 5T6
Clinton P. T. Groth
Affiliation:
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada M3H 5T6
Pierre Q. Gauthier
Affiliation:
Energy Engineering & Technology, Rolls-Royce Canada Limited, 9245 Côte-de-Liesse, Dorval, Québec, Canada H9P 1A5
*
* Corresponding author. Email addresses:[email protected] (M. R. J. Charest), [email protected] (C. P. T. Groth), [email protected] (P. Q. Gauthier)
Get access

Abstract

High-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.

Keywords

35Q3065Z0565M0865N0876M1276D0576G25Numerical algorithmscomputational fluid dynamicshigh-order methodsincompressible flows
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 3 , March 2015 , pp. 615 - 656
Copyright
Copyright © Global-Science Press 2015 

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]Drikakis, D. and Rider, W.. High-Resolution Methods for Incompressible and Low-Speed Flows. Springer-Verlag, Berlin, Heidelberg, 2005.Google Scholar
[2]Kwak, D., Kiris, C., and Housman, J.. Implicit methods for viscous incompressible flows. Comput. Fluids, 41(1):51–64, 2011.Google Scholar
[3]Patankar, S.V. and Spalding, D.B.. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer, 15(10):1787–1806, 1972.Google Scholar
[4]Rhie, C.M. and Chow, W.L.. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J., 21(11):1525–1532, 1983.Google Scholar
[5]Yanenko, N.N.. The Method of Fractional Steps. Springer-Verlag, Berlin, 1971.Google Scholar
[6]Kim, J. and Moin, P.. Application of a fractional-step method to incompressible navier-stokes equations. J. Comput. Phys., 59(2):308–323, 1985.Google Scholar
[7]Fasel, H.. Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations. J. Fluid Mech., 78(2):355–383, 1976.CrossRefGoogle Scholar
[8]Dennis, S.C.R., Ingham, D.B., and Cook, R.N.. Finite-difference methods for calculating steady incompressible flows in three dimensions. J. Comput. Phys., 33(3):325–339, 1979.CrossRefGoogle Scholar
[9]Chorin, A.J.. A numerical method for solving incompressible viscous flow problems. J. Comput. Phys., 2(1):12–26, 1967.Google Scholar
[10]Drikakis, D., Govatsos, P., and Papantonis, D.. A characteristic-based method for incompressible flows. Int. J. Numer. Meth. Fluids, 19(8):667–685, 1994.Google Scholar
[11]Shapiro, E. and Drikakis, D.. Non-conservative and conservative formulations of characteristics-based numerical reconstructions for incompressible flows. Int. J. Numer. Meth. Engin., 66(9):1466–1482, 2006.CrossRefGoogle Scholar
[12]Van Leer, B., Lee, W.T., and Roe, P.L.. Characteristic time-stepping or local preconditioning of the euler equations. AIAA Paper 91–1552, 1991.Google Scholar
[13]Choi, Y.H. and Merkle, C.L.. The application of preconditioning in viscous flows. J. Comput. Phys., 105(2):207–223, 1993.Google Scholar
[14]Turkel, E.. Review of preconditioning methods for fluid dynamics. Appl. Numer. Math., 12 (1–3):257–284, 1993.CrossRefGoogle Scholar
[15]Weiss, J.M. and Smith, W.A.. Preconditioning applied to variable and constant density flows. AIAA J., 33(11):2050–2057, 1995.Google Scholar
[16]Turkel, E., Radespiel, R., and Kroll, N.. Assessment of preconditioning methods for multidimensional aerodynamics. Comput. Fluids, 26(6):613–634, 1997.Google Scholar
[17]Venkateswaran, S., Deshpande, M., and Merkle, C.L.. The application of preconditioning to reacting flow computations. 12th AIAA Computational Fluid Dynamics Conference, San Diego, CA, Jun. 19–22 1995. AIAA paper 1995–1673.Google Scholar
[18]Steger, J.L. and Kutler, P.. Implicit finite-difference procedures for the computation of vortex wakes. AIAA J., 15(4):581–590, 1977.Google Scholar
[19]Kwak, D., Chang, J.L.C., Shanks, S.P., and Chakravarthy, S.R.. Three-dimensional incompressible Navier-Stokes flow solver using primitive variables. AIAA J., 24(3):390–396, 1986.CrossRefGoogle Scholar
[20]Turkel, E.. Preconditioned methods for solving the incompressible and low speed compressible equations. J. Comput. Phys., 72(2):277–298, 1987.CrossRefGoogle Scholar
[21]Rogers, S.E., Kwak, D., and Kaul, U.. On the accuracy of the pseudocompressibility method in solving the incompressible Navier-Stokes equations. Appl. Math. Model., 11(1):35–44, 1987a.Google Scholar
[22]Kiris, C., Kwak, D., Rogers, S., and Chang, I.D.. Computational approach for probing the flow through artificial heart devices. J. Biomed. Eng., 119(4):452–460, 1997.Google Scholar
[23]Rogers, S.E. and Kwak, D.. Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA J., 28(2):253–262, 1990.Google Scholar
[24]Rogers, S.E., Kwak, D., and Kiris, C.. Steady and unsteady solutions of the incompressible Navier-Stokes equations. AIAA J., 29(4):603–610, 1991.Google Scholar
[25]Qian, Z. and Zhang, J.. Implicit preconditioned high-order compact scheme for the simulation of the three-dimensional incompressible Navier-Stokes equations with pseudo-compressibility method. Int. J. Numer. Meth. Fluids, 69(7):1165–1185, 2012.Google Scholar
[26]Chang, J.L.C. and Kwak, D.. On the method of pseudo compressibility for numerically solving incompressible flows. AIAA Paper 84–0252, 1984.CrossRefGoogle Scholar
[27]Pirozzoli, S.. On the spectral properties of shock-capturing schemes. J. Comput. Phys., 219 (2):489–497, 2006.Google Scholar
[28]Harten, A., Engquist, B., Osher, S., and Chakravarthy, S.R.. Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys., 71(2):231–303, 1987.Google Scholar
[29]Coirier, W.J. and Powell, K.G.. Solution-adaptive Cartesian cell approach for viscous and inviscid flows. AIAA J., 34(5):938–945, May 1996.Google Scholar
[30]Barth, T.J.. Recent developments in high order k-exact reconstruction on unstructured meshes. AIAA Paper 93–0668, 1993.Google Scholar
[31]Abgrall, R.. On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys., 114:45–58, 1994.Google Scholar
[32]Sonar, T.. On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection. Comp. Meth. Appl. Mech. Eng., pp. 140–157, 1997.Google Scholar
[33]Ollivier-Gooch, C.F.. Quasi-ENO schemes for unstructured meshes based on unlimited data-dependent least-squares reconstruction. J. Comput. Phys., 133:6–17, 1997.Google Scholar
[34]Jiang, G.S. and Shu, C.W.. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126(1):202–228, 1996.Google Scholar
[35]Stanescu, D. and Habashi, W.. Essentially nonoscillatory Euler solutions on unstructured meshes using extrapolation. AIAA J., 36:1413–1416, 1998.CrossRefGoogle Scholar
[36]Friedrich, O.. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys., 144:194–212, 1998.Google Scholar
[37]Hu, C. and Shu, C.W.. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys., 150:97–127, 1999.Google Scholar
[38]Ollivier-Gooch, C.F. and Van Altena, M.. A high-order accurate unstructured mesh finite-volume scheme for the advection-diffusion equation. J. Comput. Phys., 181(2):729–752, 2002.Google Scholar
[39]Nejat, A. and Ollivier-Gooch, C.. A high-order accurate unstructured finite volume newton-krylov algorithm for inviscid compressible flows. J. Comput. Phys., 227(4):2582–2609, 2008.Google Scholar
[40]Cockburn, B. and Shu, C.W.. TVB Runge-Kutta local projection discontinous Galerkin finite-element method for conservation laws II: General framework. Math. Comp., 52:411, 1989.Google Scholar
[41]Cockburn, B., Hou, S., and Shu, C.W.. TVB Runge-Kutta local projection discontinous Galerkin finite-element method for conservation laws IV: The multidimensional case. J. Comput. Phys., 54:545, 1990.Google Scholar
[42]Hartmann, R. and Houston, P.. Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. J. Comput. Phys., 183:508–532, 2002.CrossRefGoogle Scholar
[43]Luo, H., Baum, J.D., and Löhner, R.. A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys., 225:686–713, 2007.Google Scholar
[44]Gassner, G., Lörcher, F., and Munz, C.D.. A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys., 224(2): 1049–1063, 2007.CrossRefGoogle Scholar
[45]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: 267–279, 1997.Google Scholar
[46]Cockburn, B. and Shu, C.W.. The local discontinuous Galerkin method for time-dependent convection diffusion system. SIAM J. Numer. Anal., 35(6):2440–2463, 1998.Google Scholar
[47]Leervan, B., Lo, M., and Raaltevan, M.. A discontinuous Galerkin method for diffusion based on recovery. AIAA Paper 2007–4083, 2007.Google Scholar
[48]Raaltevan, M. and Leervan, B.. Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion. Commun. Comput. Phys., 5(2–4):683–693, 2009.Google Scholar
[49]Liu, H. and Yan, J.. The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal., 41(1):675–698, 2009.Google Scholar
[50]Wang, Z.J.. Spectral (finite) volume method for conservation laws on unstructured grids – basic formulation. J. Comput. Phys., 178:210–251, 2002.Google Scholar
[51]Wang, Z.J. and Liu, Y.. Spectral (finite) volume method for conservation laws on unstructured grids – ii. extenstion to two-dimensional scalar equation. J. Comput. Phys., 179: 665–697, 2002.Google Scholar
[52]Wang, Z.J., Zhang, L., and Liu, Y.. High-order spectral volume method for 2d euler equations. Paper 2003–3534, AIAA, June 2003.Google Scholar
[53]Wang, Z.J. and Liu, Y.. Spectral (finite) volume method for conservation laws on unstructured grids – iii. one dimensional systems and partition optimization. Journal of Scientific Computing, 20(1):137–157, 2004.Google Scholar
[54]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(1):41–58, 2006.Google Scholar
[55]Huynh, H.. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper 2007–4079, 2007.Google Scholar
[56]Wang, Z.J. and Gao, H.. A unifying lifting collocation penalty formulation for the Euler equations on mixed grids. AIAA Paper 2009–401, 2009a.Google Scholar
[57]Wang, Z.J. and Gao, H.. A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys., 228:8161–8186, 2009b.CrossRefGoogle Scholar
[58]Haselbacher, A.. A WENO reconstruction algorithm for unstructed grids based on explicit stencil construction. AIAA paper 2005–0879, 2005.Google Scholar
[59]Ivan, L. and Groth, C.P.T.. High-order central CENO finite-volume scheme with adaptive mesh refinement. AIAA paper 2007–4323, 2007.Google Scholar
[60]Ivan, L. and Groth, C.P.T.. High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows. AIAA paper 2011–0367, 2011.Google Scholar
[61]Ivan, L. and Groth, C.P.T.. High-order central ENO scheme with adaptive mesh refinement for hyperbolic conservation laws. Commun. Comput. Phys., 2013a. submitted for publication.Google Scholar
[62]Ivan, L. and Groth, C.P.T.. High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows. J. Comput. Phys., 2013b. submitted for publication.CrossRefGoogle Scholar
[63]McDonald, S.D., Charest, M.R.J., and Groth, C.P.T.. High-order CENOfinite-volume schemes for multi-block unstructured mesh. 20th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, June 27–30 2011. AIAA-2011–3854.Google Scholar
[64]Susanto, A., Ivan, L., De Sterck, H., and Groth, C.P.T.. High-order central ENO finite-volume scheme for ideal MHD. J. Comput. Phys., 250(1):141 – 164, 2013.Google Scholar
[65]Rogers, S.E. and Kwak, D.. An upwind differencing scheme for the incompressible navier-strokes equations. Appl. Numer. Math., 8(1):43–64, 1991.CrossRefGoogle Scholar
[66]Chen, Y.N., Yang, S.C., and Yang, J.Y.. Implicit weighted essentially non-oscillatory schemes for the incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 31(4):747–765, 1999.Google Scholar
[67]Rogers, S.E., Chang, J.L.C., and Kwak, D.. A diagonal algorithm for the method of pseudo-compressibility. J. Comput. Phys., 73(2):364–379, 1987b.Google Scholar
[68]Yoon, S. and Kwak, D.. LU-SGS implicit algorithm for three-dimensional incompressible Navier-Stokes equations with source term. AIAA Paper 89–1964, 1989.Google Scholar
[69]Dembo, R.S., Eisenstat, S.C., and Steihaug, T.. Inexact newton methods. SIAM J. Numer. Anal., 19(2):400–408, 1982.Google Scholar
[70]Chan, T.F. and Jackson, K.R.. Nonlinearly preconditioned krylov subspace methods for discrete newton algorithms. SIAM J. Sci. Stat. Comput., 5(3):533–542, 1984.Google Scholar
[71]Brown, P.N. and Saad, Y.. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput., 11(3):450–481, 1990.Google Scholar
[72]Knoll, D.A. and Keyes, D.E.. Jacobian-free Newton-Krylov methods: A survey of approaches and applications. J. Comput. Phys., 193(2):357–397, 2004.Google Scholar
[73]Knoll, D.A., Mousseau, V.A., Chacn, L., and Reisner, J.. Jacobian-free Newton-Krylov methods for the accurate time integration of stiff wave systems. J. Comput. Phys., 25(1):213–230, 2005.Google Scholar
[74]Pan, D. and Chang, C.. The capturing of free surfaces in incompressible multi-fluid flows. Int. J. Numer. Meth. Fluids, 33(2):203–222, 2000.Google Scholar
[75]Qian, L., Causon, D.M., Ingram, D.M., and Mingham, C.G.. Cartesian cut cell two-fluid solver for hydraulic flow problems. J. Hydraul. Eng., 129(9):688–696, 2003.CrossRefGoogle Scholar
[76]Riedel, U.. A finite volume scheme on unstructured grids for stiff chemically reacting flows. Combust. Sci. Tech., 135(1–6):99–116, 1998.Google Scholar
[77]Shapiro, E. and Drikakis, D.. Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. part i. derivation of different formulations and constant density limit. J. Comput. Phys., 210(2):584–607, 2005.Google Scholar
[78]Azhdarzadeh, M. and Razavi, S.E.. A pseudo-characteristic based method for incompressible flows with heat transfer. J. Appl. Sci., 8(18):3183–3190, 2008.Google Scholar
[79]Chang, S.L. and Rhee, K.T.. Blackbody radiation functions. Int. Commun. Heat Mass Transfer, 11(5):451–455, 1984.Google Scholar
[80]Jameson, A.. Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. AIAA paper 1991–1596, 1991.Google Scholar
[81]Merkle, C.L. and Athavale, M.. Time-accurate unsteady incompressible flow algorithms based on artificial compressibility. AIAA paper 87–1137, 1987.Google Scholar
[82]Soh, W.Y. and Goodrich, J.W.. Unsteady solution of incompressible Navier-Stokes equations. J. Comput. Phys., 79(1):113–134, 1988.Google Scholar
[83]Qian, Z., Zhang, J., and Li, C.. Preconditioned pseudo-compressibility methods for incompressible navier-stokes equations. Science China: Physics, Mechanics and Astronomy, 53 (11):2090–2102, 2010.Google Scholar
[84]Lee, H. and Lee, S.. Convergence characteristics of upwind method for modified artificial compressibility method. Int. J. Aeronaut. Space Sci., 12(4):318–330, 2011.Google Scholar
[85]Malan, A.G., Lewis, R.W., and Nithiarasu, P.. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I. theory and implementation. Int. J. Numer. Meth. Engin., 54(5):695–714, 2002a.Google Scholar
[86]Malan, A.G., Lewis, R.W., and Nithiarasu, P.. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part II. application. Int. J. Numer. Meth. Engin., 54(5):715–729, 2002b.Google Scholar
[87]Naff, R.L., Russell, T.F., and Wilson, J.D.. Shape functions for three-dimensional control-volume mixed finite-element methods on irregular grids. Developments in Water Science, vol. 47, pp. 359–366. Elsevier, 2002a.Google Scholar
[88]Naff, R., Russell, T., and Wilson, J.. Shape functions for velocity interpolation in general hexahedral cells. Computat. Geosci., 6:285–314, 2002b.Google Scholar
[89]Felippa, C.A.. A compendium of FEM integration formulas for symbolic work. Eng. Computation., 21(8):867–890, 2004.Google Scholar
[90]Ivan, L.. Development of high-order CENO finite-volume schemes with block-based adaptive mesh refinement. PhD thesis, University of Toronto, 2011.Google Scholar
[91]Lawson, C.L. and Hanson, R.J.. Solving least squares problems. Prentice-Hall, 1974.Google Scholar
[92]Mavriplis, D.J.. Revisiting the least-squares procedure for gradient reconstruction on unstructured meshes. AIAA paper 2003–3986, 2003.Google Scholar
[93]Jalali, A. and Ollivier-Gooch, C.. Higher-order finite volume solution reconstruction on highly anisotropic meshes. AIAA Paper 2013–2565, 2013.Google Scholar
[94]Park, J.S., Yoon, S.H., and Kim, C.. Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J. Comput. Phys., 229(3):788–812, 2010.Google Scholar
[95]Venkatakrishnan, V.. On the accuracy of limiters and convergence to steady state solutions. AIAA Paper 93–0880, 1993.Google Scholar
[96]Roe, P.L.. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43:357–372, 1981.Google Scholar
[97]Roe, P.L. and Pike, J.. Efficient construction and utilisation of approximate Riemann solutions. Glowinski, R. and Lions, J.L., editors, Computing Methods in Applied Science and Engineering, vol. VI, pp. 499–518, Amsterdam, 1984. North-Holland.Google Scholar
[98]Mathur, S.R. and Murthy, J.Y.. A pressure-based method for unstructured meshes. Numer. Heat Transfer, Part B, 31(2):195–215, 1997.Google Scholar
[99]Karypis, G. and Schloegel, K.. ParMeTis: Parallel graph graph partitioning and sparse matrix ordering library, Version 4.0. http://www.cs.umn.edu/metis, 2011.Google Scholar
[100]Gropp, W., Lusk, E., and Skjellum, A.. Using MPI. MIT Press, Cambridge, Massachussets, 1999.Google Scholar
[101]Groth, C.P.T. and Northrup, S.A.. Parallel implicit adaptive mesh refinement scheme for body-fitted multi-block mesh. 17th AIAA Computational Fluid Dynamics Conference, Toronto, Ontario, Canada, 6–9 June 2005. AIAA paper 2005–5333.Google Scholar
[102]Charest, M.R.J., Groth, C.P.T., and Gülder, Ö.L.. A computational framework for predicting laminar reactive flows with soot formation. Combust. Theor. Modelling, 14(6):793–825, 2010.Google Scholar
[103]Charest, M.R.J., Groth, C.P.T., and Gülder, Ö.L.. Solution of the equation of radiative transfer using a Newton-Krylov approach and adaptive mesh refinement. J. Comput. Phys., 231(8): 3023–3040, 2012.Google Scholar
[104]Saad, Y. and Schultz, M.H.. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3):856–869, 1986.Google Scholar
[105]Saad, Y.. Krylov subspace methods on supercomputers. SIAM J. Sci. Stat. Comput., 10(6): 1200–1232, 1989.Google Scholar
[106]Saad, Y.. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston, 1996.Google Scholar
[107]Cuthill, E. and McKee, J.. Reducing the bandwidth of sparse symmetric matrices. Proceedings of the 1969 24th National Conference, ACM ′69, pp. 157–172, New York, NY, USA, 1969. ACM.Google Scholar
[108]Mulder, W.A. and Van Leer, B.. Experiments with implicit upwind methods for the Euler equations. J. Comput. Phys., 59:232–246, 1985.Google Scholar
[109]Bijl, H., Carpenter, M.H., and Vatsa, V.N.. Time integration schemes for the unsteady Navier-Stokes equations. AIAA Paper 2001–2612, 2001.Google Scholar
[110]Bijl, H., Carpenter, M.H., Vatsa, V.N., and Kennedy, C.A.. Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: Laminar flow. J. Comput. Phys., 179:313–329, 2002.CrossRefGoogle Scholar
[111]Lomax, H., Pulliam, T.H., and Zingg, D.W.. Fundamentals of Computational Fluid Dynamics. Springer, New York, 2003.Google Scholar
[112]Tabesh, M. and Zingg, D.W.. Efficient implicit time-marching methods using a Newton-Krylov algorithm. 47th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, 5–8 January 2009. AIAA paper 2009–0164.Google Scholar
[113]Genz, A. and Cools, R.. An adaptive numerical cubature algorithm for simplices. ACM Trans. Math. Softw., 29(3):297–308, 2003.CrossRefGoogle Scholar
[114]Cools, R. and Haegemans, A.. Algorithm 824: CUBPACK: A package for automatic cubature; framework description. ACM Trans. Math. Softw., 29(3):287–296, 2003.Google Scholar
[115]Abgrall, R.. Design of an essentially non-oscillatory reconstruction procedure on finite-element type meshes. ICASE Contractor Report 189574, 1991.Google Scholar
[116]Taylor, G.I. and Green, A.E.. Mechanism of the production of small eddies from large ones. Proc. Royal Soc. London A, 158(895):499–521, 1937.Google Scholar
[117]Shankar, P. and Deshpande, M.. Fluid mechanics in the driven cavity. Ann. Rev. Fluid Mech., 32(1):93–136, 2000.Google Scholar
[118]Jiang, B.N., Lin, T., and Povinelli, L.A.. Large-scale computation of incompressible viscous flow by least-squares finite element method. Comp. Meth. Appl. Mech. Eng., 114(3):213–231, 1994.Google Scholar
[119]Yang, J.Y., Yang, S.C., Chen, Y.N., and Hsu, C.A.. Implicit weighted ENO schemes for the three-dimensional incompressible Navier–Stokes equations. J. Comput. Phys., 146(1):464–487, 1998.Google Scholar
[120]Sachdev, J.S., Groth, C.P.T., and Gottlieb, J.J.. A parallel solution-adaptive scheme for multiphase core flows in solid propellant rocket motors. Int. J. Comput. Fluid Dyn., 19(2):159–177, 2005.Google Scholar
[121]Gao, X. and Groth, C.P.T.. A parallel adaptive mesh refinement algorithm for predicting turbulent non-premixed combusting flows. Int. J. Comput. Fluid Dyn., 20(5):349–357, 2006.Google Scholar
[122]Gao, X. and Groth, C.P.T.. A parallel solution-adaptive method for three-dimensional turbulent non-premixed combusting flows. J. Comput. Phys., 229(9):3250–3275, 2010.Google Scholar
[123]Gao, X., Northrup, S., and Groth, C.P.T.. Parallel solution-adaptive method for two-dimensional non-premixed combusting flows. Prog. Comput. Fluid. Dy., 11(2):76–95, 2011.Google Scholar