Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T22:01:50.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Facilitating the Adoption of Unstructured High-Order MethodsAmongst a Wider Community of Fluid Dynamicists

Published online by Cambridge University Press:  16 May 2011

P. E. Vincent  and
A. Jameson
Get access

Abstract

Theoretical studies and numerical experiments suggest that unstructured high-ordermethods can provide solutions to otherwise intractable fluid flow problems within complexgeometries. However, it remains the case that existing high-order schemes are generallyless robust and more complex to implement than their low-order counterparts. These issues,in conjunction with difficulties generating high-order meshes, have limited the adoptionof high-order techniques in both academia (where the use of low-order schemes remainswidespread) and industry (where the use of low-order schemes is ubiquitous). In this shortreview, issues that have hitherto prevented the use of high-order methods amongst anon-specialist community are identified, and current efforts to overcome these issues arediscussed. Attention is focused on four areas, namely the generation of unstructuredhigh-order meshes, the development of simple and efficient time integration schemes, th edevelopment of robust and accurate shock capturing algorithms, and finally the developmentof high-order methods that are intuitive and simple to implement. With regards to thisfinal area, particular attention is focused on the recently proposed flux reconstructionapproach, which allows various well known high-order schemes (such as nodal discontinuousGalerkin methods and spectral difference methods) to be cast within a single unifyingframework. It should be noted that due to the experience of the authors the review isdirected somewhat towards aerodynamic applications and compressible flow. However, many ofthe discussions have a wider applicability. Moreover, the tone of the review is intendedto be generally accessible, such that an extended scientific community can gain insightinto factors currently pacing the adoption of unstructured high-order methods.

Keywords

high-orderunstructuredmesh generationtime integrationshock capturing
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 3: Computational aerodynamics , 2011 , pp. 97 - 140
Copyright
© EDP Sciences, 2011

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

Références

R. Abgrall. On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. ICASE Report 92-74, (1992).
Anderson, W., Thomas, J., and Whitfield, D.. Multigrid acceleration of the flux-split euler equations. AIAA Journal, 26 (1988), No. 6, 649-654. CrossRefGoogle Scholar
Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D.. Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39 (2001), No. 5, 1749-1779. CrossRefGoogle Scholar
G. E. Barter and D. L. Darmofal. Shock capturing with higher-order PDE-based artificial viscosity. AIAA Paper 2007-3823, 2007.
Barter, G. E. and Darmofal, D. L.. Shock capturing with PDE-based artificial viscosity for DGFEM. J. Comput. Phys., 229 (2010), No. 5, 1810-1827. CrossRefGoogle Scholar
T. J. Barth and H. Deconinck. High-order methods for computational physics. Springer Verlag, 1999.
T. J. Barth and P. O. Frederickson. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper 90-0013, 1990.
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), No. 2, 267-279. CrossRefGoogle Scholar
Bassi, F. and Rebay, S.. High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys., 138 (1997), No. 2, 251-285. CrossRefGoogle Scholar
Bhagatwala, A. and Lele, S. K.. A modified artificial viscosity approach for compressible turbulence simulations. J. Comput. Phys., 228 (2009), 4965-4969. CrossRefGoogle Scholar
Brooks, A. N. and Hughes, T. J. R.. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Method. Appl. M., 32 (1982), 199-259. CrossRefGoogle Scholar
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods: Fundamentals in single domains. Springer, 2006.
P. Castonguay, C. Liang, and A. Jameson. Simulation of transitional flow over airfoils using the spectral difference method. AIAA Paper 2010-4626, 2010.
Caughey, D.. Diagonal implicit multigrid algorithm for the Euler equations. AIAA Journal, 26 (1988), 841-851. CrossRefGoogle Scholar
Chen, R. F. and Wang, Z. J.. Fast, block lower-upper symmetric Gauss-Seidel scheme for arbitrary grids. AIAA Journal, 38 (2000) 2238-2245. CrossRefGoogle Scholar
Cockburn, B., Gopalakrishnan, J., and Lazarov, R.. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 47 (2009), No. 2, 1319-1365. CrossRefGoogle Scholar
Cockburn, B., Hou, S., and Shu, C. W.. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comput., 54 (1990), 545-581. Google Scholar
B. Cockburn, G. E. Karniadakis, and C. W. Shu. Discontinuous Galerkin methods: Theory, computation and applications. Springer, 2000.
Cockburn, B., Lin, S. Y., and Shu, C. W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems. J. Comput. Phys., 84 (1989), No. 1, 90-113. CrossRefGoogle Scholar
Cockburn, B. and Shu, C.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comput., 52 (1989), 411-435. Google Scholar
Cockburn, B. and Shu, C. W.. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal., 35 (1998), No. 6, 2440-2463. CrossRefGoogle Scholar
Cockburn, B. and Shu, C. W.. The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. J. Comput. Phys., 141 (1998), No. 2, 199-224. CrossRefGoogle Scholar
Cockburn, B. and Shu, C. W.. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16 (2001), No. 3, 173-261. CrossRefGoogle Scholar
Cook, A. W.. Artificial fluid properties for large-eddy simulation of compressible turbulent mixing. Phys. Fluids, 19 (2007), 55-103. CrossRefGoogle Scholar
Cook, A. W. and Cabot, W. H.. A high-wavenumber viscosity for high-resolution numerical methods. J. Comput. Phys., 195 (2004), No. 2, 594-601. CrossRefGoogle Scholar
Cook, A. W. and Cabot, W. H.. Hyperviscosity for shock-turbulence interactions. J. Comput. Phys., 203 (2005), 379-385. CrossRefGoogle Scholar
J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs. Isogeometric analysis: Toward integration of CAD and FEA. Wiley, 2009.
M. Delanaye and Y. Liu. uadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA Paper 1999-3259, 1999.
M. O. Deville, P. F. Fischer, and E. H. Mund. High-order methods for incompressible fluid flow. Cambridge University Press, 2002.
S. Dey, R. M. O’bara, and M. S. Shephard. Curvilinear mesh generation in 3D. In Proceedings of the Eighth International Meshing Roundtable, John Wiley & Sons, (1999) 407-417.
Dolean, V., Fahs, H., Fezoui, L., and Lanteri, S.. Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys., 229 (2010), No 2, 512-526. CrossRefGoogle Scholar
J. Douglas and T. Dupont. Interior penalty procedures for elliptic and parabolic Galerkin methods. In Computing Methods in Applied Sciences (Second International Symposium, Versailles, 1975), Springer, (1976), 207-216.
Dubief, Y. and Delcayre, F.. On coherent-vortex identification in turbulence. J. Turbul., 1 (2000), No. 11, 1-22. CrossRefGoogle Scholar
Dumbser, M., Balsara, D. S., Toro, E. F., and Munz, C. D.. A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys., 227 (2008), 8209-8253. CrossRefGoogle Scholar
J. K. Fidkowski and D. L. Darmofal. Output-based error estimation and mesh adaptation in computational fluid dynamics: Overview and recent results. AIAA Paper 2009-1303, 2009.
Fidkowski, K. J., Oliver, T. A., Lu, J., and Darmofal, D. L.. p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys., 207 (2005), No. 1, 92-113. CrossRefGoogle Scholar
Fidkowski, K. J. and Roe, P. L.. An entropy adjoint approach to mesh refinement. SIAM J Sci Comput, 32 (2010), No. 3, 261-1287. CrossRefGoogle Scholar
Friedrich, O.. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys., 144 (1998), No. 1, 194-212. CrossRefGoogle Scholar
M. Galbraith and M. Visbal. Implicit large eddy simulation of low Reynolds number flow past the SD7003 airfoil. AIAA Paper 2008-225, 2008.
Gao, H., Wang, Z. J., and Liu, Y.. A study of curved boundary representations for 2D high order Euler solvers. J. Sci. Comput., 44 (2010), 323-336. CrossRefGoogle Scholar
Gassner, G., Lorcher, F., and Munz, C. D.. A discontinuous Galerkin scheme based on a space-time expansion II: Viscous flow equations in multi dimensions. J. Sci. Comput., 34 (2008), No. 3, 260-286. CrossRefGoogle Scholar
Geuzaine, C. and Remacle, J.. Gmsh: A 3D finite element mesh generator with built-in pre and post-processing facilities. Int. J. Numer. Meth. Eng., 79 (2009), No. 11, 1309-1331. CrossRefGoogle Scholar
Gottlieb, S., Shu, C. W., and Tadmor, E.. Strong stability-preserving high-order time discretization methods. SIAM Review, 43 (2001), No. 1, 89-112. CrossRefGoogle Scholar
T. Haga, H. Gao, and Z. J. Wang. A high-order unifying discontinuous formulation for 3D mixed grids. AIAA Paper 2010-540, 2010.
Haga, T., Sawada, K., and Wang, Z. J.. An implicit LU-SGS scheme for the spectral volume method on unstructured tetrahedral grids. Commun. Comput. Phys., 6 (2009), No. 5, 978-996. CrossRefGoogle Scholar
Harten, A.. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49 (1983), No. 3, 357-393. CrossRefGoogle Scholar
Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. R.. Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys., 72 (1987), No. 2, 231-303. CrossRefGoogle Scholar
Hartmann, R.. Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids, 51 (2006), 1131-1156. CrossRefGoogle Scholar
Helenbrook, B. T. and Atkins, H. L.. Solving discontinuous Galerkin formulations of Poisson’s equation using geometric and p-multigrid. AIAA Journal, 46 (2008), No. 4, 894-916. CrossRefGoogle Scholar
Hesthaven, J. S. and Gottlieb, D.. Stable spectral methods for conservation laws on triangles with unstructured grids. Comput. Method Appl. M., 175 (1999), 361-381. CrossRefGoogle Scholar
J. S. Hesthaven and T. Warburton. Nodal discontinuous Galerkin methods - Algorithms, analysis, and applications. Springer, 2008.
Hu, C. and Shu, C. W.. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys., 150 (1999), No. 1, 97-127. CrossRefGoogle Scholar
T. J. R. Hughes and A. N. Brooks. A multidimensional upwind scheme with no crosswind diffusion. In T. J. R. Hughes, editor, Finite element methods for convection dominated flows, ASME, New York, (1979), 19-35.
T. J. R. Hughes and A. N. Brooks. A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: Application to the streamline upwind procedure. In R. H. Gallagher, D. H. Norrie, J. T. Oden, and O. C. Zienkiewicz, editors, Finite elements in fluids, volume IV, Wiley, London, (1982), 46-65.
Hughes, T. J. R., Cottrell, J. A., and Bazilevs, Y.. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Method Appl. M., 194 (2005), 4135-4195. CrossRefGoogle Scholar
Hughes, T. J. R. and Mallet, M.. A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems. Comput. Method Appl. M., 58 (1986), 305-328. CrossRefGoogle Scholar
Hughes, T. J. R., Mallet, M., and Mizukami, A.. A new finite element formulation for computational fluid dynamics: II. beyond SUPG. Comput. Method Appl. M., 54 (1986), 341-355. CrossRefGoogle Scholar
H. T. Huynh. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper 2007-4079, 2007.
H. T. Huynh. A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion. AIAA Paper 2009-403, 2009.
F. Iacono and G. May. Convergence acceleration for simulation of steady-state compressible flows using high-order schemes. AIAA Paper 2009-4132, 2009.
F. Iacono, G. May, and Z. J. Wang. Relaxation techniques for high-order discretizations of steady compressible inviscid flows. AIAA Paper 2010-4991, 2010.
Jameson, A.. Solution of the Euler equations for two dimensional transonic flow by a multigrid method. Appl. Math. Comput., 13 (1983), No. 3, 327-356. Google Scholar
Jameson, A.. A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput., 45 (2010), 348-358. CrossRefGoogle Scholar
A. Jameson and T. J. Baker. Solution of the Euler equations for complex configurations. AIAA Paper 83-1929, 1983.
A. Jameson and D. A. Caughey. How many steps are required to solve the Euler equations of steady, compressible flow: In search of a fast solution algorithm. AIAA Paper 2001-2673, 2001.
A. Jameson, W. Schmidt, and E. Turkel. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA Paper 81-1259, 1981.
Jameson, A. and Yoon, S.. Multigrid solution of the Euler equations using implicit schemes. AIAA Journal, 24 (1986), No. 11, 1737-1743. CrossRefGoogle Scholar
Jameson, A. and Yoon, S.. Lower-upper implicit schemes with multiple grids for the Euler equations. AIAA Journal, 25 (1987), No. 7, 929-935. CrossRefGoogle Scholar
Jones, K. D., Dohring, C. M., and Platzer, M. F.. Experimental and computational investigation of the Knoller-Betz effect. AIAA Journal, 36 (1998), 780-783. CrossRefGoogle Scholar
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 (2009), No. 2, 165-199. CrossRefGoogle Scholar
Karamanos, G. S. and Karniadakis, G. E.. A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys., 163 (2000), No. 2, 22-50. CrossRefGoogle Scholar
G. E. Karniadakis and S. J. Sherwin. Spectral/hp element methods for computational fluid dynamics. Oxford Scientific Publications, 2nd edition, 2005.
Kawai, S. and Lele, S. K.. Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J. Comput. Phys., 227 (2008), No. 22, 9498-9526. CrossRefGoogle Scholar
Kirby, R. M. and Sherwin, S. J.. Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling. Comput. Method Appl. M., 195 (2006), 3128-3144. CrossRefGoogle Scholar
Klöckner, A., Warburton, T., Bridge, J., and Hesthaven, J. S.. Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys., 228 (2009), No. 21, 7863-7882. CrossRefGoogle Scholar
Kopriva, D. A. and Kolias, J. H.. A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys., 125 (1996), No. 1, 244-261. CrossRefGoogle Scholar
R. J. Leveque. Finite volume methods for hyperbolic problems. Cambridge University Press, 2002.
Y. Li, S. Premasuthan, and A. Jameson. Comparison of h and p-adaptations for spectral difference methods. AIAA Paper 2010-4435, 2010.
Liang, C., Jameson, A., and Wang, Z. J.. Spectral difference method for compressible flow on unstructured grids with mixed elements. J. Comput. Phys., 228 (2009), No. 8, 2847-2858. CrossRefGoogle Scholar
Liang, C., Kannan, R., and Wang, Z. J.. A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids. Comput. Fluids, 38 (2009), No. 2, 254-265. CrossRefGoogle Scholar
Liu, L., Li, X., and Hu, F. Q.. Nonuniform time-step Runge-Kutta discontinuous Galerkin method for computational aeroacoustics. J. Comput. Phys., 229 (2010), 6874-6897. CrossRefGoogle Scholar
Liu, X. D., Osher, S., and Chan, T.. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115 (1994), No. 1, 200-212. CrossRefGoogle Scholar
Liu, Y., Vinokur, M., and Wang, Z. J.. Spectral difference method for unstructured grids I: Basic formulation. J. Comput. Phys., 216 (2006), No. 2, 780-801. CrossRefGoogle Scholar
Lorcher, F., Gassner, G., and Munz, C. D.. A discontinuous Galerkin scheme based on a space-time expansion I. Inviscid compressible flow in one space dimension. J. Sci. Comput., 32 (2007), No. 2, 175-199. CrossRefGoogle Scholar
Luo, H., Baum, J. D., and Lohner, R.. A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids. J. Comput. Phys., 211 (2006), No. 2, 767-783. CrossRefGoogle Scholar
Maday, Y. and Munoz, R.. Spectral element multigrid II. Theoretical justification. J. Sci. Comput., 3 (1988), No. 4, 323-353. CrossRefGoogle Scholar
Mascarenhas, B. S., Helenbrook, B. T., and Atkins, H. L.. Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection-diffusion equation. J. Comput. Phys., 229 (2010), No. 10, 3664-3674. CrossRefGoogle Scholar
D. J. Mavriplis and C. R. Nastase. On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes. AIAA Paper 2008-778, 2008.
G. May. The spectral difference scheme as a quadrature-free discontinuous Galerkin method. Aachen Institute for Advanced Study Technical Report AICES-2008-11, 2008.
G. May and A. Jameson. Efficient relaxation methods for high-order discretization of steady problems. In Adaptive high-order methods in computational fluid dynamics (advances in computational fluid dynamics). In Press.
Nastase, C. R. and Mavriplis, D. J.. High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys., 213 (2006), No. 2, 330-357. CrossRefGoogle Scholar
Nguyen, N. C., Peraire, J., and Cockburn, B.. An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. J. Comput. Phys., 228 (2009), No. 9, 3232-3254. CrossRefGoogle Scholar
Nguyen, N. C., Peraire, J., and Cockburn, B.. An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys., 228 (2009), No. 23, 8841-8855. CrossRefGoogle Scholar
N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations. J. Comput. Phys., In press, 2010.
Ollivier-Gooch, C. F.. Quasi-ENO schemes for unstructured meshes based on unlimited data-dependent least-squares reconstruction. J. Comput. Phys., 133 (1997), No. 1, 6-17. CrossRefGoogle Scholar
K. Ou and A. Jameson. A high-order spectral difference method for fluid-structure interaction on dynamic deforming meshes. AIAA Paper 2010-4932, 2010.
K. Ou, C. Liang, and A. Jameson. A high-order spectral difference method for the Navier-Stokes equations on unstructured moving deformable grids. AIAA Paper 2010-541, 2010.
K. Ou, C. Liang, S. Premasuthan, and A. Jameson. High-order spectral difference simulation of laminar compressible flow over two counter-rotating cylinders. AIAA Paper 2009-3956, 2009.
Parsani, M., van den Abeele, K., Lacor, C., and Turkel, E.. Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier-Stokes equations. J. Comput. Phys., 229 (2010), No. 3, 828-850. CrossRefGoogle Scholar
Peraire, J. and Persson, P.. The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput., 30 (2008), No. 4, 1806-1824. CrossRefGoogle Scholar
Persson, P., Bonet, J., and Peraire, J.. Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains. Comput. Method Appl. M., 198 (2009), 1585-1595. CrossRefGoogle Scholar
P. Persson and J. Peraire. Sub-cell shock capturing for discontinuous Galerkin methods. AIAA Paper 2006-112, 2006.
Persson, P. and Peraire, J.. Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations. SIAM J. Sci. Comput., 30 (2008), No. 6, 2709-2722. CrossRefGoogle Scholar
P. Persson and J. Peraire. Curved mesh generation and mesh refinement using Lagrangian solid mechanics. AIAA Paper 2009-949, 2009.
P. Persson, D. J. Willis, and J. Peraire. The numerical simulation of flapping wings at low Reynolds numbers. AIAA Paper 2010-724, 2010.
S. Premasuthan, C. Liang, and A. Jameson. A spectral difference method for viscous compressible flows with shocks. AIAA Paper 2009-3785, June 2009.
S. Premasuthan, C. Liang, and A. Jameson. Computation of flows with shocks using spectral difference scheme with artificial viscosity. AIAA Paper 2010-1449, 2010.
S. Premasuthan, C. Liang, A. Jameson, and Z. J. Wang. A p-multigrid spectral difference method for viscous compressible flow using 2D quadrilateral meshes. AIAA Paper 2009-950, 2009.
Qiu, J. and Shu, C. W.. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput., 26 (2005), No. 3, 907-929. CrossRefGoogle Scholar
Radespiel, R., Windte, J., and Scholz, U.. Numerical and experimental flow analysis of moving airfoils with laminar separation bubbles. AIAA Journal, 45 (2007), 1346-1356. CrossRefGoogle Scholar
W. H. Reed and T. R. Hill. T riangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos National Laboratory, New Mexico, USA, 1973.
Roe, P. L.. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43 (1981), No. 2, 357-372. CrossRefGoogle Scholar
Ronquist, E. M. and Patera, A. T.. Spectral element multigrid I. Formulation and numerical results. J. Sci. Comput., 2 (1987), No. 4, 389-406. CrossRefGoogle Scholar
Sevilla, R., Fernandez-Mendez, S., and Huerta, A.. NURBS-enhanced finite element method for Euler equations. Int. J. Numer. Meth. Fluids, 57 (2008), No. 9, 1051-1069. CrossRefGoogle Scholar
Sevilla, R., Fernandez-Mendez, S., and Huerta, A.. NURBS-enhanced finite element method (NEFEM). Int. J. Numer. Meth. Engng., 76 (2008), No. 1, 56-83. CrossRefGoogle Scholar
Sherwin, S. J. and Ainsworth, M.. Unsteady Navier-Stokes solvers using hybrid spectral/hp element methods. Appl. Numer. Math., 33 (2000), No. 1, 357-364. CrossRefGoogle Scholar
Sherwin, S. J. and Karniadakis, G. E.. A new triangular and tetrahedral basis for high-order (hp) finite element methods. Int. J. Numer. Meth. Eng., 38 (1995), No. 22, 3775-3802. CrossRefGoogle Scholar
Sherwin, S. J. and Peiro, J.. Mesh generation in curvilinear domains using high-order elements. Int. J. Numer. Meth. Eng., 53 (2002), No. 1, 207-223. CrossRefGoogle Scholar
Sherwin, S. J., Warburton, T. C. E., and Karniadakis, G. E.. Spectral/hp methods for elliptic problems on hybrid grids. Contemp. Math., 218 (1998), 191-216. CrossRefGoogle Scholar
Shu, C. W.. Total-variation-diminishing time discretizations. SIAM J. Sci. and Stat. Comput., 9 (1988), No. 6, 1073-1084. CrossRefGoogle Scholar
Shu, C. W. and Osher, S.. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77 (1988), No. 2, 439-471. CrossRefGoogle Scholar
J. C. Strikwerda. Finite difference schemes and partial differential equations. SIAM, 2nd edition, 2004.
Sun, Y., Wang, Z. J., and Liu, Y.. High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Commun. Comput. Phys., 2 (2007), 310-333. Google Scholar
Sun, Y., Wang, Z. J., and Liu, Y.. Efficient implicit non-linear LU-SGS approach for compressible flow computation using high-order spectral difference method. Commun. Comput. Phys., 5 (2009), 760-778. Google Scholar
Tadmor, E.. Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal., 26 (1989), No. 1, 30-44. CrossRefGoogle Scholar
A. Uranga, P. Persson, M. Drela, and J. Peraire. Implicit large eddy simulation of transitional flows over airfoils and wings. AIAA Paper 2009-4131, 2009.
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 (2007), No. 2, 616-636. CrossRefGoogle Scholar
Van den Abeele, K., Lacor, C., and Wang, Z. J.. On the connection between the spectral volume and the spectral difference method. J. Comput. Phys., 227 (2007), No. 2, 877-885. CrossRefGoogle Scholar
Van den Abeele, K., Lacor, C., and Wang, Z. J.. On the stability and accuracy of the spectral difference method. J. Sci. Comput., 37 (2008), No. 2, 162-188. CrossRefGoogle Scholar
B. van Leer. Towards the ultimate conservative difference scheme I. The quest of monotonicity. In Proceedings of the third international conference on numerical methods in fluid mechanics, Springer, (1973), 163-168.
van Leer, B.. Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys., 14 (1974), No. 4, 361-370. CrossRefGoogle Scholar
van Leer, B.. Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow. J. Comput. Phys., 23 (1977), No. 3, 263-275. CrossRefGoogle Scholar
van Leer, B.. Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Comput. Phys., 23 (1977), No. 3, 276-299. CrossRefGoogle Scholar
van Leer, B.. Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method. J. Comput. Phys., 32 (1979), No. 1, 101-136. CrossRefGoogle Scholar
Venkatakrishnan, V.. Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J. Comput. Phys., 118 (1995), No. 1, 120-130. CrossRefGoogle Scholar
P. E. Vincent, P. Castonguay, and A. Jameson. A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput., (2010), In press.
von Neumann, J. and Richtmyer, R. D.. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 21 (1950), 232-237. CrossRefGoogle Scholar
Wang, Z. J.. Spectral (finite) volume method for conservation laws on unstructured grids: Basic formulation. J. Comput. Phys., 178 (2002), No. 2, 210-251. CrossRefGoogle Scholar
Wang, Z. J.. High-order methods for the Euler and Navier-Stokes equations on unstructured grids. Prog. Aerosp. Sci., 43 (2007), 1-41. CrossRefGoogle Scholar
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 (2009), No. 21, 8161-8186. CrossRefGoogle Scholar
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., 179 (2002), No. 2, 665-697. CrossRefGoogle Scholar
Wang, Z. J. and Liu, Y.. Spectral (finite) volume method for conservation laws on unstructured grids III: One dimensional systems and partition optimization. J. Sci. Comput., 20 (2004), No. 1, 137-157. CrossRefGoogle Scholar
Wang, Z. J., Zhang, L., and Liu, Y.. Spectral (finite) volume method for conservation laws on unstructured grids IV: Extension to two-dimensional systems. J. Comput. Phys., 194 (2004), No. 2, 716-741. CrossRefGoogle Scholar
A. Wolkov, Ch. Hirsch, and B. Leonard. Discontinuous Galerkin method on unstructured hexahedral grids for 3D Euler and Navier-Stokes equations. AIAA Paper 2007-4078, 2007.
Yoon, S. and Jameson, A.. Lower-upper symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations. AIAA Journal, 26 (1988), No. 9, 1025-1026. CrossRefGoogle Scholar
Zhang, Y., Bazilevs, Y., Goswami, S., Bajaj, C. L., and Hughes, T. J. R.. Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput. Method Appl. M., 196 (2007), 2943-2959. CrossRefGoogle ScholarPubMed
Y. Zhou and Z. J. Wang. Implicit large eddy simulation of transitional flow over a SD7003 wing using high-order spectral difference method. AIAA Paper 2010-4442, 2010.
Zhu, J., Qiu, J., Shu, C. W., and Dumbser, M.. Runge-Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes. J. Comput. Phys., 227 (2008), No. 9, 4330-4353. CrossRefGoogle Scholar
O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu. The finite element method Its basis and fundamentals. Elsevier, 6th edition, 2005.