Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T09:23:32.674Z Has data issue: false hasContentIssue false
  • English
  • Français

Application of a Higher Order DiscontinuousGalerkin

Published online by Cambridge University Press:  16 May 2011

A. V. Wolkov ,
Ch. Hirsch  and
N. B. Petrovskaya
A. V. Wolkov*
Affiliation:
Central Aerohydrodynamic Institute, Zhukovsky, Moscow Region, 140180, Russia
Ch. Hirsch
Affiliation:
Vrije Universiteit Brussel, Belgium
N. B. Petrovskaya
Affiliation:
University of Birmingham, B15 2TT, Birmingham, UK
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

We discuss the issues of implementation of a higher order discontinuous Galerkin (DG)scheme for aerodynamics computations. In recent years a DG method has intensively beenstudied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has beendesigned for numerical solution of the 3-D Euler and Navier-Stokes equations. Ourdiscussion is mainly based on the results of the DG study conducted in TsAGI incollaboration with the NUMECA International. The capacity of a DG scheme to tacklechallenging computational problems is demonstrated and its potential advantages over FVschemes widely used in modern computational aerodynamics are highlighted.

Keywords

computational aerodynamicsdiscontinuous Galerkinhigher order schemes
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 3: Computational aerodynamics , 2011 , pp. 237 - 263
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

J. D. Anderson, Jr. Fundamentals of aerodynamics. McGraw-Hill, New York, 1991.
Agarwal, R. K., Halt, D. W.. A Compact high-order unstructured grids method for the solution of Euler equations. Int. J. Num.Meth. Fluids, 31 (1999), 121–147. 3.0.CO;2-S>CrossRefGoogle Scholar
Barth, T. J.. Numerical methods for gasdynamic systems on unstructured meshes. Lecture Notes in Comput. Sci. Engrg., 8 (1998), 195–284. Google Scholar
T. Barth, P. Frederickson. Higher-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA 90-0013, 1990.
G. E. Barter, D. L. Darmofal. Shock capturing with high-order,PDE-based artificial viscosity. AIAA paper 2007-3823, 2007.
F. Bassi, A. Crivellini, D. A. Di Pietro, S. Rebay. A high-order discontinuous Galerkin solver for 3D aerodynamic turbulent flows. in Proceedings of ECCOMAS CFD 2006, P. Wesseling, E. Onate and J. Periaux (Eds), 2006
Bassi, F., Rebay, S.. Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids, 40 (2002), No. 1, 197–207. CrossRefGoogle Scholar
Burbeau, A., Sagaut, P., Bruneau, Ch.-H.. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. J.Comput.Phys., 169 (2001), 111–150. CrossRefGoogle Scholar
J. C. Butcher. Numerical methods for ordinary differential equations. John Wiley & Sons, 2003.
Casoni, E., Peraire, J., Huerta, A.. One-dimensional shock-capturing for high-order discontinuous Galerkin methods. Computational Methods in Applied Sciences, 14 (2009), 307–325. Google Scholar
Cockburn, B.. Discontinuous Galerkin methods for convection - dominated problems. Lecture Notes in Comput. Sci. Engrg., 9 (1999), 69–224. CrossRefGoogle Scholar
Cockburn, B., Karniadakis, G. E., Shu, C.-W.. The development of discontinuous Galerkin methods. Lecture Notes in Comput. Sci. Engrg., 9 (2000), 3–50. CrossRefGoogle Scholar
Cockburn, B., Shu, C.-W.. The local discontinuous Galerkin method for time-dependent convection-diffusion system. SIAM. J. Numer. Anal., 35 (1998), 2440–2463. CrossRefGoogle Scholar
Cockburn, B., Shu, C.-W.. The Runge - Kutta discontinuous Galerkin method for conservation laws V. J. Comput. Phys., 141 (1998), 199–224. CrossRefGoogle Scholar
P. H. Cook, M. A. McDonald, M. C. P. Firmin. Aerofoil RAE 2822 – pressure distribution, and boundary layer and wake measurements. AGARD-AR-138.
D. L. Darmofal, R. Haimes. Towards the next generation in CFD. AIAA 2005-0087, 2005.
M. Delanaye, A. Patel, B. Leonard, Ch. Hirsch. Automatic unstructured hexahedral grid generation and flow solution. in Proceedings of ECCOMAS CFD-2001, Swansea, Wales, UK, 2001.
Dolejší, V., Feistauer, M., Schwab, C.. On some aspects of the discontinuous Galerkin finite element method for conservation laws. Mathematics and Computers in Simulation, 61 (2003), 333–346. CrossRefGoogle Scholar
Enayet, M. M., Gibson, M. M., Taylor, A. M. K. P., Yianneskis, M.. Laser-Doppler measurements of laminar and turbulent flow in a pipe bend. Int J. Heat and Fluid Flow, 3 (1982), No. 4, 213–219. CrossRefGoogle Scholar
Engquist, B., Osher, S.. One-sided difference equations for nonlinear conservation laws. Math. Comp., 36 (1981), 321–352. CrossRefGoogle Scholar
C. Hirsch. Numerical computation of internal and external flows. vol.2, John Wiley & Sons, 1990.
H. Hoteit et al. New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes. INRIA report No. 4491, INRIA Rennes, France, 2002.
F. Q. Hu, M. Y. Hussaini, J. Manthey. Low-dissipation and -dispersion Runge-Kutta schemes for computational acoustics. NASA Technical Report, 1994.
Kershaw, D. S., Prasad, M. K., Shaw, M. J., Milovich, J. L.. 3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method. Comput. Meth. Appl. Mech. Engrg., 158 (1998), 81–116. CrossRefGoogle Scholar
Krivodonova, L.. Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys., 226 (2007), No. 1, 276–296. CrossRefGoogle Scholar
Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., Flaherty, J. E.. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Num. Math., 48 (2004), 323–338. CrossRefGoogle Scholar
A. G. Kulikovskii, N. V. Pogorelov, A. Yu. Semenov. Mathematical aspects of numerical solution of hyperbolic systems. Monographs and Surveys in Pure and Applied Mathematics, 188, Chapman and Hall/CRC, Boca Raton, Florida, 2001.
E. M. Lee-Rausch, P. G. Buning, D. Mavriplis, J. H. Morrison, M. A. Park, S. M. Rivers, C. L. Rumsey. CFD sensitivity analysis of a Drag Prediction Workshop wing/body transport configuration. AIAA 2003-3400, 2003.
R. J. LeVeque. Numerical methods for conservation laws. Birkhäuser Verlag, Basel, Switzerland, 1992.
Levy, D. W., Zickuhr, T., Vassberg, J., Agrawal, S., Wahls, R. A., Pirzadeh, S., Hemsh, M. J.. Data summary from the first AIAA Computational Fluid Dynamics Drag Prediction Workshop. J. Aircraft, 40 (2003), No. 5, 875–882. CrossRefGoogle Scholar
R. B. Lowrier. Compact higher-order numerical methods for hyperbolic conservation laws. PhD thesis, The University of Michigan, 1996.
Luo, H., Baum, J. D., Löhner, R.. A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys., 225 (2007), 686–713. CrossRefGoogle Scholar
Luo, H., Baum, J. D., Löhner, R.. A fast, p-multigrid discontinuous Galerkin method for compressible flows at all speeds. AIAA Journal, 46 (2008), No. 3, 635–652. CrossRefGoogle Scholar
A. A. Martynov, S. Yu. Medvedev. A robust method of anisotropic grid generation. In:Grid generation: Theory and Applications, Computing Centre RAS, Moscow, (2002), 266-275.
D. J. Mavriplis. Unstructured mesh discretizations and solvers for computational aerodynamics. AIAA 2007-3955, 2007.
C. R. Nastase, D. J. Mavriplis. Discontinuous Galerkin methods using an hp-multigrid solver for inviscid compressible flows on three-dimensional unstructured meshes. AIAA-Paper 2006-107, 2006.
P.-O. Persson, J. Peraire. Sub-cell shock capturing for discontinuous Galerkin method. AIAA paper 2006-112, 2006.
Petrovskaya, N. B., Wolkov, A. V.. The issues of solution approximation in higher order schemes on distorted grids. Int. J. Comput. Methods, 4 (2007), No. 2, 367–382. CrossRefGoogle Scholar
Petrovskaya, N. B.. Quadratic least-squares solution reconstruction in a boundary layer region. Commun. Numer. Meth. Engng., 26 (2010), No. 12, 1721–1735. Google Scholar
Petrovskaya, N. B.. Discontinuous weighted least-squares approximation on irregular grids. CMES: Computer Modeling in Engineering & Sciences, 32 (2008), No. 2, 69–84. Google Scholar
Petrovskaya, N. B., Wolkov, A. V., Lyapunov, S. V.. Modification of basis functions in high order discontinuous Galerkin schemes for advection equation. Appl. Math. Mod., 32 (2008), No. 5, 826–835. CrossRefGoogle Scholar
Qiu, J., Shu, C.-W.. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys., 193 (2003), 115–135. CrossRefGoogle Scholar
Shu, C.-W., Osher, S.. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77 (1988), 439-471. CrossRefGoogle Scholar
Spalart, P. R., Allmaras, S. R.. A one-equation turbulence model for aerodynamic flows. La Recherche Aérospatiale, 1 (1994), 5–21. Google Scholar
Y. Sun, Z. J. Wang. Evaluation of discontinuous Galerkin and spectral volume methods for conservation laws on unstructured grids, AIAA 2003-0253, 2003.
Tam, C. K. W., Webb, J. C.. Dispersion-relation-preserving schemes for computational acoustics, J. Comput. Phys., 107 (1993), 262–281. CrossRefGoogle Scholar
J. J. W. van der Vegt, H. van der Ven. Space – time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flow. 33rd Computational Fluid Dynamics Course ‘Novel methods for solving convection dominated systems’, the von Karman Institute, Rhode-St-Genese, Belgium, March 24–28, 2003.
V. Venkatakrishnan, S. Allmaras, D. Kamenetskii, F. Johnson. Higher order schemes for the compressible Navier-Stokes equations. AIAA 2003-3987, 2003.
A. V. Wolkov. Design and implementation of higher order schemes for 3-D computational aerodynamics problems. Habilitation Thesis, Central Aerohydrodynamic Institute (TsAGI), Moscow, 2010.
Wolkov, A. V.. Application of the multigrid approach for solving the 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method. J. Comput. Mathem. and Mathem. Phys., 50 (2010), No. 3, 495–508. CrossRefGoogle Scholar
A. Wolkov, Ch. Hirsch, B. Leonard. Discontinuous Galerkin method on unstructured hexahedral grids for the 3D Euler and Navier-Stokes equations. AIAA 2007-4078, 2007.
A. Wolkov, Ch. Hirsch, B. Leonard. Discontinuous Galerkin method on unstructured hexahedral grids. AIAA 2009-177, 2009.
Wolkov, A. V., Petrovskaya, N. B.. Higher order discontinuous Galerkin method for acoustic pulse problem. Comput. Phys. Commun., 181 (2010), 1186–1194. CrossRefGoogle Scholar
J. Zhu, J. Qiu, C.-W. Shu , Dumbser, M.. Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes. J. Comput. Phys., 227 (2008), 4330–4353. Google Scholar