# Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Mathematical Modelling of Natural Phenomena (2011)

- Volume: 6, Issue: 3, page 97-140
- ISSN: 0973-5348

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topVincent, P. E., and Jameson, A.. "Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists." Mathematical Modelling of Natural Phenomena 6.3 (2011): 97-140. <http://eudml.org/doc/222361>.

@article{Vincent2011,

abstract = {Theoretical studies and numerical experiments suggest that unstructured high-order
methods can provide solutions to otherwise intractable fluid flow problems within complex
geometries. However, it remains the case that existing high-order schemes are generally
less robust and more complex to implement than their low-order counterparts. These issues,
in conjunction with difficulties generating high-order meshes, have limited the adoption
of high-order techniques in both academia (where the use of low-order schemes remains
widespread) and industry (where the use of low-order schemes is ubiquitous). In this short
review, issues that have hitherto prevented the use of high-order methods amongst a
non-specialist community are identified, and current efforts to overcome these issues are
discussed. Attention is focused on four areas, namely the generation of unstructured
high-order meshes, the development of simple and efficient time integration schemes, th e
development of robust and accurate shock capturing algorithms, and finally the development
of high-order methods that are intuitive and simple to implement. With regards to this
final area, particular attention is focused on the recently proposed flux reconstruction
approach, which allows various well known high-order schemes (such as nodal discontinuous
Galerkin methods and spectral difference methods) to be cast within a single unifying
framework. It should be noted that due to the experience of the authors the review is
directed somewhat towards aerodynamic applications and compressible flow. However, many of
the discussions have a wider applicability. Moreover, the tone of the review is intended
to be generally accessible, such that an extended scientific community can gain insight
into factors currently pacing the adoption of unstructured high-order methods. },

author = {Vincent, P. E., Jameson, A.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {high-order; unstructured; mesh generation; time integration; shock capturing},

language = {eng},

month = {5},

number = {3},

pages = {97-140},

publisher = {EDP Sciences},

title = {Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists},

url = {http://eudml.org/doc/222361},

volume = {6},

year = {2011},

}

TY - JOUR

AU - Vincent, P. E.

AU - Jameson, A.

TI - Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

JO - Mathematical Modelling of Natural Phenomena

DA - 2011/5//

PB - EDP Sciences

VL - 6

IS - 3

SP - 97

EP - 140

AB - Theoretical studies and numerical experiments suggest that unstructured high-order
methods can provide solutions to otherwise intractable fluid flow problems within complex
geometries. However, it remains the case that existing high-order schemes are generally
less robust and more complex to implement than their low-order counterparts. These issues,
in conjunction with difficulties generating high-order meshes, have limited the adoption
of high-order techniques in both academia (where the use of low-order schemes remains
widespread) and industry (where the use of low-order schemes is ubiquitous). In this short
review, issues that have hitherto prevented the use of high-order methods amongst a
non-specialist community are identified, and current efforts to overcome these issues are
discussed. Attention is focused on four areas, namely the generation of unstructured
high-order meshes, the development of simple and efficient time integration schemes, th e
development of robust and accurate shock capturing algorithms, and finally the development
of high-order methods that are intuitive and simple to implement. With regards to this
final area, particular attention is focused on the recently proposed flux reconstruction
approach, which allows various well known high-order schemes (such as nodal discontinuous
Galerkin methods and spectral difference methods) to be cast within a single unifying
framework. It should be noted that due to the experience of the authors the review is
directed somewhat towards aerodynamic applications and compressible flow. However, many of
the discussions have a wider applicability. Moreover, the tone of the review is intended
to be generally accessible, such that an extended scientific community can gain insight
into factors currently pacing the adoption of unstructured high-order methods.

LA - eng

KW - high-order; unstructured; mesh generation; time integration; shock capturing

UR - http://eudml.org/doc/222361

ER -

## References

top- R. Abgrall. On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. ICASE Report 92-74, (1992).
- W. Anderson, J. Thomas, and D. Whitfield. Multigrid acceleration of the flux-split euler equations. AIAA Journal, 26 (1988), No. 6, 649-654.
- D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39 (2001), No. 5, 1749-1779.
- G. E. Barter and D. L. Darmofal. Shock capturing with higher-order PDE-based artificial viscosity. AIAA Paper 2007-3823, 2007.
- G. E. Barter and D. L. Darmofal. Shock capturing with PDE-based artificial viscosity for DGFEM. J. Comput. Phys., 229 (2010), No. 5, 1810-1827.
- 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.
- F. Bassi and S. Rebay. 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.
- F. Bassi and S. Rebay. High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys., 138 (1997), No. 2, 251-285.
- A. Bhagatwala and S. K. Lele. A modified artificial viscosity approach for compressible turbulence simulations. J. Comput. Phys., 228 (2009), 4965-4969.
- A. N. Brooks and T. J. R. Hughes. 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.
- 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.
- D. Caughey. Diagonal implicit multigrid algorithm for the Euler equations. AIAA Journal, 26 (1988), 841-851.
- R. F. Chen and Z. J. Wang. Fast, block lower-upper symmetric Gauss-Seidel scheme for arbitrary grids. AIAA Journal, 38 (2000) 2238-2245.
- B. Cockburn, J. Gopalakrishnan, and R. Lazarov. 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.
- B. Cockburn, S. Hou, and C. W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comput., 54 (1990), 545-581.
- B. Cockburn, G. E. Karniadakis, and C. W. Shu. Discontinuous Galerkin methods: Theory, computation and applications. Springer, 2000.
- B. Cockburn, S. Y. Lin, and C. W. Shu. 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.
- B. Cockburn and C. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comput., 52 (1989), 411-435.
- B. Cockburn and C. W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal., 35 (1998), No. 6, 2440-2463.
- B. Cockburn and C. W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. J. Comput. Phys., 141 (1998), No. 2, 199-224.
- B. Cockburn and C. W. Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16 (2001), No. 3, 173-261.
- A. W. Cook. Artificial fluid properties for large-eddy simulation of compressible turbulent mixing. Phys. Fluids, 19 (2007), 55-103.
- A. W. Cook and W. H. Cabot. A high-wavenumber viscosity for high-resolution numerical methods. J. Comput. Phys., 195 (2004), No. 2, 594-601.
- A. W. Cook and W. H. Cabot. Hyperviscosity for shock-turbulence interactions. J. Comput. Phys., 203 (2005), 379-385.
- 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.
- V. Dolean, H. Fahs, L. Fezoui, and S. Lanteri. Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys., 229 (2010), No 2, 512-526.
- 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.
- Y. Dubief and F. Delcayre. On coherent-vortex identification in turbulence. J. Turbul., 1 (2000), No. 11, 1-22.
- M. Dumbser, D. S. Balsara, E. F. Toro, and C. D. Munz. 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.
- 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.
- K. J. Fidkowski, T. A. Oliver, J. Lu, and D. L. Darmofal. p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys., 207 (2005), No. 1, 92-113.
- K. J. Fidkowski and P. L. Roe. An entropy adjoint approach to mesh refinement. SIAM J Sci Comput, 32 (2010), No. 3, 261-1287.
- O. Friedrich. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys., 144 (1998), No. 1, 194-212.
- M. Galbraith and M. Visbal. Implicit large eddy simulation of low Reynolds number flow past the SD7003 airfoil. AIAA Paper 2008-225, 2008.
- H. Gao, Z. J. Wang, and Y. Liu. A study of curved boundary representations for 2D high order Euler solvers. J. Sci. Comput., 44 (2010), 323-336.
- G. Gassner, F. Lorcher, and C. D. Munz. 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.
- C. Geuzaine and J. Remacle. 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.
- S. Gottlieb, C. W. Shu, and E. Tadmor. Strong stability-preserving high-order time discretization methods. SIAM Review, 43 (2001), No. 1, 89-112.
- T. Haga, H. Gao, and Z. J. Wang. A high-order unifying discontinuous formulation for 3D mixed grids. AIAA Paper 2010-540, 2010.
- T. Haga, K. Sawada, and Z. J. Wang. An implicit LU-SGS scheme for the spectral volume method on unstructured tetrahedral grids. Commun. Comput. Phys., 6 (2009), No. 5, 978-996.
- A. Harten. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49 (1983), No. 3, 357-393.
- A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy. Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys., 72 (1987), No. 2, 231-303.
- R. Hartmann. Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids, 51 (2006), 1131-1156.
- B. T. Helenbrook and H. L. Atkins. Solving discontinuous Galerkin formulations of Poisson’s equation using geometric and p-multigrid. AIAA Journal, 46 (2008), No. 4, 894-916.
- J. S. Hesthaven and D. Gottlieb. Stable spectral methods for conservation laws on triangles with unstructured grids. Comput. Method Appl. M., 175 (1999), 361-381.
- J. S. Hesthaven and T. Warburton. Nodal discontinuous Galerkin methods - Algorithms, analysis, and applications. Springer, 2008.
- C. Hu and C. W. Shu. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys., 150 (1999), No. 1, 97-127.
- 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.
- T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Method Appl. M., 194 (2005), 4135-4195.
- T. J. R. Hughes and M. Mallet. 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.
- T. J. R. Hughes, M. Mallet, and A. Mizukami. A new finite element formulation for computational fluid dynamics: II. beyond SUPG. Comput. Method Appl. M., 54 (1986), 341-355.
- 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.
- A. Jameson. Solution of the Euler equations for two dimensional transonic flow by a multigrid method. Appl. Math. Comput., 13 (1983), No. 3, 327-356.
- A. Jameson. A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput., 45 (2010), 348-358.
- 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.
- A. Jameson and S. Yoon. Multigrid solution of the Euler equations using implicit schemes. AIAA Journal, 24 (1986), No. 11, 1737-1743.
- A. Jameson and S. Yoon. Lower-upper implicit schemes with multiple grids for the Euler equations. AIAA Journal, 25 (1987), No. 7, 929-935.
- K. D. Jones, C. M. Dohring, and M. F. Platzer. Experimental and computational investigation of the Knoller-Betz effect. AIAA Journal, 36 (1998), 780-783.
- R. Kannan and Z. J. Wang. A study of viscous flux formulations for a p-multigrid spectral volume Navier-Stokes solver. J. Sci. Comput., 41 (2009), No. 2, 165-199.
- G. S. Karamanos and G. E. Karniadakis. A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys., 163 (2000), No. 2, 22-50.
- G. E. Karniadakis and S. J. Sherwin. Spectral/hp element methods for computational fluid dynamics. Oxford Scientific Publications, 2nd edition, 2005.
- S. Kawai and S. K. Lele. Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J. Comput. Phys., 227 (2008), No. 22, 9498-9526.
- R. M. Kirby and S. J. Sherwin. Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling. Comput. Method Appl. M., 195 (2006), 3128-3144.
- A. Klöckner, T. Warburton, J. Bridge, and J. S. Hesthaven. Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys., 228 (2009), No. 21, 7863-7882.
- D. A. Kopriva and J. H. Kolias. A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys., 125 (1996), No. 1, 244-261.
- 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.
- C. Liang, A. Jameson, and Z. J. Wang. Spectral difference method for compressible flow on unstructured grids with mixed elements. J. Comput. Phys., 228 (2009), No. 8, 2847-2858.
- C. Liang, R. Kannan, and Z. J. Wang. A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids. Comput. Fluids, 38 (2009), No. 2, 254-265.
- L. Liu, X. Li, and F. Q. Hu. Nonuniform time-step Runge-Kutta discontinuous Galerkin method for computational aeroacoustics. J. Comput. Phys., 229 (2010), 6874-6897.
- X. D. Liu, S. Osher, and T. Chan. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115 (1994), No. 1, 200-212.
- Y. Liu, M. Vinokur, and Z. J. Wang. Spectral difference method for unstructured grids I: Basic formulation. J. Comput. Phys., 216 (2006), No. 2, 780-801.
- F. Lorcher, G. Gassner, and C. D. Munz. 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.
- H. Luo, J. D. Baum, and R. Lohner. A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids. J. Comput. Phys., 211 (2006), No. 2, 767-783.
- Y. Maday and R. Munoz. Spectral element multigrid II. Theoretical justification. J. Sci. Comput., 3 (1988), No. 4, 323-353.
- B. S. Mascarenhas, B. T. Helenbrook, and H. L. Atkins. Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection-diffusion equation. J. Comput. Phys., 229 (2010), No. 10, 3664-3674.
- 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.
- C. R. Nastase and D. J. Mavriplis. High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys., 213 (2006), No. 2, 330-357.
- N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. J. Comput. Phys., 228 (2009), No. 9, 3232-3254.
- N. C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys., 228 (2009), No. 23, 8841-8855.
- 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.
- C. F. Ollivier-Gooch. Quasi-ENO schemes for unstructured meshes based on unlimited data-dependent least-squares reconstruction. J. Comput. Phys., 133 (1997), No. 1, 6-17.
- 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.
- M. Parsani, K. van den Abeele, C. Lacor, and E. Turkel. 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.
- J. Peraire and P. Persson. The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput., 30 (2008), No. 4, 1806-1824.
- P. Persson, J. Bonet, and J. Peraire. Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains. Comput. Method Appl. M., 198 (2009), 1585-1595.
- P. Persson and J. Peraire. Sub-cell shock capturing for discontinuous Galerkin methods. AIAA Paper 2006-112, 2006.
- P. Persson and J. Peraire. Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations. SIAM J. Sci. Comput., 30 (2008), No. 6, 2709-2722.
- 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.
- J. Qiu and C. W. Shu. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput., 26 (2005), No. 3, 907-929.
- R. Radespiel, J. Windte, and U. Scholz. Numerical and experimental flow analysis of moving airfoils with laminar separation bubbles. AIAA Journal, 45 (2007), 1346-1356.
- 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.
- P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43 (1981), No. 2, 357-372.
- E. M. Ronquist and A. T. Patera. Spectral element multigrid I. Formulation and numerical results. J. Sci. Comput., 2 (1987), No. 4, 389-406.
- R. Sevilla, S. Fernandez-Mendez, and A. Huerta. NURBS-enhanced finite element method for Euler equations. Int. J. Numer. Meth. Fluids, 57 (2008), No. 9, 1051-1069.
- R. Sevilla, S. Fernandez-Mendez, and A. Huerta. NURBS-enhanced finite element method (NEFEM). Int. J. Numer. Meth. Engng., 76 (2008), No. 1, 56-83.
- S. J. Sherwin and M. Ainsworth. Unsteady Navier-Stokes solvers using hybrid spectral/hp element methods. Appl. Numer. Math., 33 (2000), No. 1, 357-364.
- S. J. Sherwin and G. E. Karniadakis. A new triangular and tetrahedral basis for high-order (hp) finite element methods. Int. J. Numer. Meth. Eng., 38 (1995), No. 22, 3775-3802.
- S. J. Sherwin and J. Peiro. Mesh generation in curvilinear domains using high-order elements. Int. J. Numer. Meth. Eng., 53 (2002), No. 1, 207-223.
- S. J. Sherwin, T. C. E. Warburton, and G. E. Karniadakis. Spectral/hp methods for elliptic problems on hybrid grids. Contemp. Math., 218 (1998), 191-216.
- C. W. Shu. Total-variation-diminishing time discretizations. SIAM J. Sci. and Stat. Comput., 9 (1988), No. 6, 1073-1084.
- C. W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77 (1988), No. 2, 439-471.
- J. C. Strikwerda. Finite difference schemes and partial differential equations. SIAM, 2nd edition, 2004.
- Y. Sun, Z. J. Wang, and Y. Liu. High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids. Commun. Comput. Phys., 2 (2007), 310-333.
- Y. Sun, Z. J. Wang, and Y. Liu. Efficient implicit non-linear LU-SGS approach for compressible flow computation using high-order spectral difference method. Commun. Comput. Phys., 5 (2009), 760-778.
- E. Tadmor. Convergence of spectral methods for nonlinear conservation laws. SIAM J. Numer. Anal., 26 (1989), No. 1, 30-44.
- 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.
- K. Van den Abeele, T. Broeckhoven, and C. Lacor. 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.
- K. Van den Abeele, C. Lacor, and Z. J. Wang. On the connection between the spectral volume and the spectral difference method. J. Comput. Phys., 227 (2007), No. 2, 877-885.
- K. Van den Abeele, C. Lacor, and Z. J. Wang. On the stability and accuracy of the spectral difference method. J. Sci. Comput., 37 (2008), No. 2, 162-188.
- 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.
- B. van Leer. 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.
- B. van Leer. 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.
- B. van Leer. Towards the ultimate conservative difference scheme IV. A new approach to numerical convection. J. Comput. Phys., 23 (1977), No. 3, 276-299.
- B. van Leer. Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method. J. Comput. Phys., 32 (1979), No. 1, 101-136.
- V. Venkatakrishnan. Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J. Comput. Phys., 118 (1995), No. 1, 120-130.
- 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.
- J. von Neumann and R. D. Richtmyer. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 21 (1950), 232-237.
- Z. J. Wang. Spectral (finite) volume method for conservation laws on unstructured grids: Basic formulation. J. Comput. Phys., 178 (2002), No. 2, 210-251.
- Z. J. Wang. High-order methods for the Euler and Navier-Stokes equations on unstructured grids. Prog. Aerosp. Sci., 43 (2007), 1-41.
- Z. J. Wang and H. Gao. 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.
- Z. J. Wang and Y. Liu. 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.
- Z. J. Wang and Y. Liu. 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.
- Z. J. Wang, L. Zhang, and Y. Liu. 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.
- 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.
- S. Yoon and A. Jameson. Lower-upper symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations. AIAA Journal, 26 (1988), No. 9, 1025-1026.
- Y. Zhang, Y. Bazilevs, S. Goswami, C. L. Bajaj, and T. J. R. Hughes. Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput. Method Appl. M., 196 (2007), 2943-2959.
- 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.
- J. Zhu, J. Qiu, C. W. Shu, and M. Dumbser. Runge-Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes. J. Comput. Phys., 227 (2008), No. 9, 4330-4353.
- O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu. The finite element method Its basis and fundamentals. Elsevier, 6th edition, 2005.

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