Application of a Higher Order Discontinuous Galerkin

A. V. Wolkov; Ch. Hirsch; N. B. Petrovskaya

Mathematical Modelling of Natural Phenomena (2011)

  • Volume: 6, Issue: 3, page 237-263
  • ISSN: 0973-5348

Abstract

top
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 been studied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has been designed for numerical solution of the 3-D Euler and Navier-Stokes equations. Our discussion is mainly based on the results of the DG study conducted in TsAGI in collaboration with the NUMECA International. The capacity of a DG scheme to tackle challenging computational problems is demonstrated and its potential advantages over FV schemes widely used in modern computational aerodynamics are highlighted.

How to cite

top

Wolkov, A. V., Hirsch, Ch., and Petrovskaya, N. B.. "Application of a Higher Order Discontinuous Galerkin." Mathematical Modelling of Natural Phenomena 6.3 (2011): 237-263. <http://eudml.org/doc/222321>.

@article{Wolkov2011,
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 been studied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has been designed for numerical solution of the 3-D Euler and Navier-Stokes equations. Our discussion is mainly based on the results of the DG study conducted in TsAGI in collaboration with the NUMECA International. The capacity of a DG scheme to tackle challenging computational problems is demonstrated and its potential advantages over FV schemes widely used in modern computational aerodynamics are highlighted. },
author = {Wolkov, A. V., Hirsch, Ch., Petrovskaya, N. B.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {computational aerodynamics; discontinuous Galerkin; higher order schemes},
language = {eng},
month = {5},
number = {3},
pages = {237-263},
publisher = {EDP Sciences},
title = {Application of a Higher Order Discontinuous Galerkin},
url = {http://eudml.org/doc/222321},
volume = {6},
year = {2011},
}

TY - JOUR
AU - Wolkov, A. V.
AU - Hirsch, Ch.
AU - Petrovskaya, N. B.
TI - Application of a Higher Order Discontinuous Galerkin
JO - Mathematical Modelling of Natural Phenomena
DA - 2011/5//
PB - EDP Sciences
VL - 6
IS - 3
SP - 237
EP - 263
AB - 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 been studied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has been designed for numerical solution of the 3-D Euler and Navier-Stokes equations. Our discussion is mainly based on the results of the DG study conducted in TsAGI in collaboration with the NUMECA International. The capacity of a DG scheme to tackle challenging computational problems is demonstrated and its potential advantages over FV schemes widely used in modern computational aerodynamics are highlighted.
LA - eng
KW - computational aerodynamics; discontinuous Galerkin; higher order schemes
UR - http://eudml.org/doc/222321
ER -

References

top
  1. J. D. Anderson, Jr. Fundamentals of aerodynamics. McGraw-Hill, New York, 1991.  
  2. R. K. Agarwal, D. W. Halt. A Compact high-order unstructured grids method for the solution of Euler equations. Int. J. Num.Meth. Fluids, 31 (1999), 121–147.  
  3. T. J. Barth. Numerical methods for gasdynamic systems on unstructured meshes. Lecture Notes in Comput. Sci. Engrg., 8 (1998), 195–284.  
  4. T. Barth, P. Frederickson. Higher-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA 90-0013, 1990.  
  5. G. E. Barter, D. L. Darmofal. Shock capturing with high-order,PDE-based artificial viscosity. AIAA paper 2007-3823, 2007.  
  6. 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  
  7. F. Bassi, S. Rebay. Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids, 40 (2002), No. 1, 197–207.  
  8. A. Burbeau, P. Sagaut, Ch.-H. Bruneau. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. J.Comput.Phys., 169 (2001), 111–150.  
  9. J. C. Butcher. Numerical methods for ordinary differential equations. John Wiley & Sons, 2003.  
  10. E. Casoni, J. Peraire, A. Huerta. One-dimensional shock-capturing for high-order discontinuous Galerkin methods. Computational Methods in Applied Sciences, 14 (2009), 307–325.  
  11. B. Cockburn. Discontinuous Galerkin methods for convection - dominated problems. Lecture Notes in Comput. Sci. Engrg., 9 (1999), 69–224.  
  12. B. Cockburn, G. E. Karniadakis, C.-W. Shu. The development of discontinuous Galerkin methods. Lecture Notes in Comput. Sci. Engrg., 9 (2000), 3–50.  
  13. B. Cockburn, C.-W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion system. SIAM. J. Numer. Anal., 35 (1998), 2440–2463.  
  14. B. Cockburn, C.-W. Shu. The Runge - Kutta discontinuous Galerkin method for conservation laws V. J. Comput. Phys., 141 (1998), 199–224.  
  15. P. H. Cook, M. A. McDonald, M. C. P. Firmin. Aerofoil RAE 2822 – pressure distribution, and boundary layer and wake measurements. AGARD-AR-138.  
  16. D. L. Darmofal, R. Haimes. Towards the next generation in CFD. AIAA 2005-0087, 2005.  
  17. 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.  
  18. V. Dolejší, M. Feistauer, C. Schwab. On some aspects of the discontinuous Galerkin finite element method for conservation laws. Mathematics and Computers in Simulation, 61 (2003), 333–346.  
  19. M. M. Enayet, M. M. Gibson, A. M. K. P. Taylor, M. Yianneskis. Laser-Doppler measurements of laminar and turbulent flow in a pipe bend. Int J. Heat and Fluid Flow, 3 (1982), No. 4, 213–219.  
  20. B. Engquist, S. Osher. One-sided difference equations for nonlinear conservation laws. Math. Comp., 36 (1981), 321–352.  
  21. C. Hirsch. Numerical computation of internal and external flows. vol.2, John Wiley & Sons, 1990.  
  22. H. Hoteit et al. New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes. INRIA report No. 4491, INRIA Rennes, France, 2002.  
  23. F. Q. Hu, M. Y. Hussaini, J. Manthey. Low-dissipation and -dispersion Runge-Kutta schemes for computational acoustics. NASA Technical Report, 1994.  
  24. D. S. Kershaw, M. K. Prasad, M. J. Shaw, J. L. Milovich. 3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method. Comput. Meth. Appl. Mech. Engrg., 158 (1998), 81–116.  
  25. L. Krivodonova. Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys., 226 (2007), No. 1, 276–296.  
  26. L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J. E. Flaherty. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Num. Math., 48 (2004), 323–338.  
  27. 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.  
  28. 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.  
  29. R. J. LeVeque. Numerical methods for conservation laws. Birkhäuser Verlag, Basel, Switzerland, 1992.  
  30. D. W. Levy, T. Zickuhr, J. Vassberg, S. Agrawal, R. A. Wahls, S. Pirzadeh, M. J. Hemsh. Data summary from the first AIAA Computational Fluid Dynamics Drag Prediction Workshop. J. Aircraft, 40 (2003), No. 5, 875–882.  
  31. R. B. Lowrier. Compact higher-order numerical methods for hyperbolic conservation laws. PhD thesis, The University of Michigan, 1996.  
  32. H. Luo, J. D. Baum, R. Löhner. A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys., 225 (2007), 686–713.  
  33. H. Luo, J. D. Baum, R. Löhner. A fast, p-multigrid discontinuous Galerkin method for compressible flows at all speeds. AIAA Journal, 46 (2008), No. 3, 635–652.  
  34. 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.  
  35. D. J. Mavriplis. Unstructured mesh discretizations and solvers for computational aerodynamics. AIAA 2007-3955, 2007.  
  36. 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.  
  37. P.-O. Persson, J. Peraire. Sub-cell shock capturing for discontinuous Galerkin method. AIAA paper 2006-112, 2006.  
  38. N. B. Petrovskaya, A. V. Wolkov. The issues of solution approximation in higher order schemes on distorted grids. Int. J. Comput. Methods, 4 (2007), No. 2, 367–382.  
  39. N. B. Petrovskaya. Quadratic least-squares solution reconstruction in a boundary layer region. Commun. Numer. Meth. Engng., 26 (2010), No. 12, 1721–1735.  
  40. N. B. Petrovskaya. Discontinuous weighted least-squares approximation on irregular grids. CMES: Computer Modeling in Engineering & Sciences, 32 (2008), No. 2, 69–84.  
  41. N. B. Petrovskaya, A. V. Wolkov, S. V. Lyapunov. Modification of basis functions in high order discontinuous Galerkin schemes for advection equation. Appl. Math. Mod., 32 (2008), No. 5, 826–835.  
  42. J. Qiu, C.-W. Shu. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys., 193 (2003), 115–135.  
  43. C.-W. Shu, S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77 (1988), 439-471.  
  44. P. R. Spalart, S. R. Allmaras. A one-equation turbulence model for aerodynamic flows. La Recherche Aérospatiale, 1 (1994), 5–21.  
  45. Y. Sun, Z. J. Wang. Evaluation of discontinuous Galerkin and spectral volume methods for conservation laws on unstructured grids, AIAA 2003-0253, 2003.  
  46. C. K. W. Tam, J. C. Webb. Dispersion-relation-preserving schemes for computational acoustics, J. Comput. Phys., 107 (1993), 262–281.  
  47. 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.  
  48. V. Venkatakrishnan, S. Allmaras, D. Kamenetskii, F. Johnson. Higher order schemes for the compressible Navier-Stokes equations. AIAA 2003-3987, 2003.  
  49. A. V. Wolkov. Design and implementation of higher order schemes for 3-D computational aerodynamics problems. Habilitation Thesis, Central Aerohydrodynamic Institute (TsAGI), Moscow, 2010.  
  50. A. V. Wolkov. 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.  
  51. 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.  
  52. A. Wolkov, Ch. Hirsch, B. Leonard. Discontinuous Galerkin method on unstructured hexahedral grids. AIAA 2009-177, 2009.  
  53. A. V. Wolkov, N. B. Petrovskaya. Higher order discontinuous Galerkin method for acoustic pulse problem. Comput. Phys. Commun., 181 (2010), 1186–1194.  
  54. J. Zhu, J. Qiu, C.-W. Shu , M. Dumbser. Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes. J. Comput. Phys., 227 (2008), 4330–4353.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.