Skip to main content
SpringerLink
Log in
Book cover

High-Order Methods for Computational Physics pp 69–224Cite as

Discontinuous Galerkin Methods for Convection-Dominated Problems

  • Chapter

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 9))

Abstract

We present and analyze the Runge Kutta Discontinuous Galerkin method for numerically solving nonlinear hyperbolic systems. The basic method is then extended to convection-dominated problems yielding the Local Discontinuous Galerkin method. These methods are particularly attractive since they achieve formal high-order 0accuracy, nonlinear stability, and high parallelizability while maintaining the ability to handle complicated geometries and capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The discussed methods are readily applied to the Euler equations of gas dynamics, the shallow water equations, the equations of magneto-hydrodynamics, the compressible Navier-Stokes equations with high Reynolds numbers, and the equations of the hydrodynamic model for semiconductor device simulation. As a final example, consideration is given to the application of the discontinuous Galerkin method to the Hamilton-Jacobi equations.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.L. Atkins and C.-W. Shu. Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations. Technical Report 96–51, ICASE, 1996. To appear in AIAA J.

    Google Scholar 

  2. I. Babuska, C.E. Baumann, and J.T. Oden. A discontinuous hp finite element method for diffusion problems: 1-D analysis. Technical Report 22, TICAM, 1997.

    Google Scholar 

  3. 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: 267–279, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  4. F. Bassi and S. Rebay. High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys., 138: 251–285, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Bassi, S. Rebay, M. Savini, G. Mariotti, and S. Pedinotti. A high-order accurate discontinuous finite element method for inviscid and viscous turbo-machinery flows. In Proceedings of the Second European Conference on Turbo-machinery Fluid Dynamics and Thermodynamics, 1997. Antwerpen, Belgium.

    Google Scholar 

  6. C.E. Baumann An hp-adaptive discontinuous Galerkin method for computational fluid dynamics. PhD thesis, The University of Texas at Austin, 1997.

    Google Scholar 

  7. C.E. Baumann and J.T. Oden. A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. To appear.

    Google Scholar 

  8. C.E. Baumann and J.T. Oden. A discontinuous hp finite element method for the Navier-Stokes equations. In 10th. International Conference on Finite Element in Fluids, 1998.

    Google Scholar 

  9. C.E. Baumann and J.T. Oden. A discontinuous hp finite element method for the solution of the Euler equation of gas dynamics. In 10th. International Conference on Finite Element in Fluids, 1998.

    Google Scholar 

  10. K.S. Bey and J.T. Oden. A Runge-Kutta discontinuous Galerkin finite element method for high speed flows. 10th. AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, June 24–27, 1991.

    Google Scholar 

  11. R. Biswas, K.D. Devine, and J. Flaherty. Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math., 14: 255–283, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Chavent and B. Cockburn. The local projection P° P1-discontinuousGalerkin finite element method for scalar conservation laws. M2 AN, 23: 565–592, 1989.

    Google Scholar 

  13. G. Chavent and G. Salzano. A finite element method for the 1D water flooding problem with gravity. J. Comput. Phys., 45: 307–344, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  14. Z. Chen, B. Cockburn, C. Gardner, and J. Jerome. Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode. J. Comput. Phys., 117: 274–280, 1995.

    Article  MATH  Google Scholar 

  15. Z. Chen, B. Cockburn, J. Jerome, and C.-W. Shu. Mixed-RKDG finite element method for the drift-diffusion semiconductor device equations. VLSI Design, 3: 145–158, 1995.

    Article  Google Scholar 

  16. P. Ciarlet. The finite element method for elliptic problems. North Holland, 1975.

    Google Scholar 

  17. B. Cockburn. An introduction to the discontinuous Galerkin method for convection-dominated problems. In Advanced numerical approximation of nonlinear hyperbolic equations, A. Quarteroni, editor, Lecture Notes in Mathematics, CIME subseries. Springer Verlag. To appear.

    Google Scholar 

  18. B. Cockburn and P.-A. Gremaud. A priori error estimates for numerical methods for scalar conservation laws. part I: The general approach. Math. Comp., 65: 533–573, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Cockburn and P.A. Gremaud. Error estimates for finite element methods for nonlinear conservation laws. SIAM J. Numer. Anal., 33: 522–554, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Cockburn, S. Hou, and C.W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comp., 54: 545–581, 1990.

    MathSciNet  MATH  Google Scholar 

  21. 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: 90–113, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Cockburn, M. Luskin, C.-W. Shu, and E. Süli. A priori error estimates for the discontinuous Galerkin method. in preparation.

    Google Scholar 

  23. B. Cockburn and C. Schwab. hp-error analysis for the local discontinuous Galerkin method. In preparation.

    Google Scholar 

  24. B. Cockburn and C.W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework. Math. Comp., 52: 411–435, 1989.

    MathSciNet  MATH  Google Scholar 

  25. B. Cockburn and C.W. Shu. The P1-RKDG method for two-dimensional Euler equations of gas dynamics. Technical Report 91–32, ICASE, 1991.

    Google Scholar 

  26. B. Cockburn and C.W. Shu. The Runge-Kutta local projection P1- discontinuous Galerkin method for scalar conservation laws. M2 AN, 25: 337–361, 1991.

    MathSciNet  MATH  Google Scholar 

  27. B. Cockburn and C.W. Shu. The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J. Numer. Anal., 35: 2440–2463, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  28. B. Cockburn and C.W. Shu. The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems. J. Comput. Phys., 141: 199–224, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  29. M.G. Crandall and H. Ishiiand P.L. Lions. User’s guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc., 27: 1–67, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  30. H.L. deCougny, K.D. Devine, J.E. Flaherty, R.M. Loy, C. Ozturan, and M.S. Shephard. Load balancing for the parallel adaptive solution of partial differential equations. Appl. Numer. Math., 16: 157–182, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  31. K.D. Devine and J.E. Flaherty. Parallel adaptive hp-refinement techniques for conservation laws. Appl. Numer. Math., 20: 367–386, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  32. K.D. Devine, J.E. Flaherty, R.M. Loy, and S.R. Wheat. Parallel partitioning strategies for the adaptive solution of conservation laws. In I Babus’ka, W.D. Henshaw, J.E. Hoperoft, J.E. Oliger, and T. Tezduyar, editors, Modeling, mesh generation, and adaptive numerical methods for partial differential equations, volume 75, pages 215–242, 1995.

    Google Scholar 

  33. K.D. Devine, J.E. Flaherty, S.R. Wheat, and A.B. Maccabe. A massively parallel adaptive finite element method with dynamic load balancing. In Proceedings Supercomputing’93, pages 2–11, 1993.

    Google Scholar 

  34. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal., 28: 43–77, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  35. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems II: Optimal error estimates in 1,12 and looloo. SIAM J. Numer. Anal., 32: 706–740, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  36. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems IV: A nonlinear model problem. SIAM J. Numer. Anal., 32: 1729–1749, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  37. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems V: Long time integration. SIAM J. Numer. Anal., 32: 1750–1762, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  38. K. Eriksson, C. Johnson, and V. Thomée. Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO, Anal. Numér., 19: 611–643, 1985.

    MATH  Google Scholar 

  39. R.S. Falk and G.R. Richter. Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. To appear.

    Google Scholar 

  40. J.E. Flaherty, R.M. Loy, M.S. Shephard, B.K. Szymanski, J.D. Teresco, and L.H. Ziantz. Adaptive refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws. Technical report, IMA Preprint Series # 1483, 1997.

    Google Scholar 

  41. J. Goodman and R. LeVeque. On the accuracy of stable schemes for 2D scalar conservation laws. Math. Comp., 45: 15–21, 1985.

    MathSciNet  MATH  Google Scholar 

  42. P. Houston, C. Schwab, and E. Süli. Stabilized hp-finite element methods for hyperbolic problems. SIAM J. Numer. Anal. To appear.

    Google Scholar 

  43. C. Hu and C.-W. Shu. A discontinuous Galerkin finite element method for Hamilton-Jacobi equations. SIAM J. Sci. Comput. To appear.

    Google Scholar 

  44. T. Hughes and A. Brook. Streamline upwind-Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg., 32: 199–259, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  45. T. Hughes, L.P. Franca, M. Mallet, and A. Misukami. A new finite element formulation for computational fluid dynamics, I. Comput. Methods Appl. Mech. Engrg., 54: 223–234, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Hughes, L.P. Franca, M. Mallet, and A. Misukami A new finite element formulation for computational fluid dynamics, II. Comput. Methods Appl. Mech. Engrg., 54: 341–355, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  47. T. Hughes, L.P. Franca, M. Mallet, and A. Misukami. A new finite element formulation for computational fluid dynamics, III. Comput. Methods Appl. Mech. Engrg., 58: 305–328, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  48. T. Hughes, L.P. Franca, M. Mallet, and A. Misukami A new finite element formulation for computational fluid dynamics, IV. Comput. Methods Appl. Mech. Engrg., 58: 329–336, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  49. T. Hughes and M. Mallet. A high-precision finite element method for shock-tube calculations. Finite Element in Fluids, 6: 339-, 1985.

    Google Scholar 

  50. P. Jamet. Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal., 15: 912–928, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  51. G. Jiang and C.-W. Shu. On cell entropy inequality for discontinuous Galerkin methods. Math. Comp., 62: 531–538, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  52. C. Johnson and J. Pitkaranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46: 1–26, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  53. C. Johnson and J. Saranen. Streamline diffusion methods for problems in fluid mechanics. Math. Comp., 47: 1–18, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  54. C. Johnson and A. Szepessy. On the convergence of a finite element method for a non-linear hyperbolic conservation law. Math. Comp., 49: 427–444, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  55. C. Johnson, A. Szepessy, and P. Hansbo. On the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comp., 54: 107–129, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  56. D.S. Kershaw, M.K. Prasad, and M.J. Shawand J.L. Milovich. 3D unstructured mesh ALE hydrodynamics with the upwind discontinuous Galerkin method. Comput. Methods Appl. Mech. Engrg., 158: 81–116, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  57. D.A. Kopriva. A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations. Technical Report 97–66, Florida State UniversitySCRI, 1997.

    Google Scholar 

  58. P. LeSaint and P.A. Raviart. On a finite element method for solving the neutron transport equation. In C. de Boor, editor, Mathematical aspects of finite elements in partial differential equations, pages 89–145. Academic Press, 1974.

    Google Scholar 

  59. Q. Lin, N. Yan, and A.-H. Zhou. An optimal error estimate of the discontinuous Galerkin method. Journal of Engineering Mathematics, 13: 101–105, 1996.

    MathSciNet  MATH  Google Scholar 

  60. Q. Lin and A.-H. Zhou. Convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. Acta Math. Sci., 13: 207–210, 1993.

    MathSciNet  MATH  Google Scholar 

  61. W. B. Lindquist. Construction of solutions for two-dimensional Riemann problems. Comp. ê9 Maths. with Appls., 12: 615–630, 1986.

    MathSciNet  MATH  Google Scholar 

  62. W. B. Lindquist. The scalar Riemann problem in two spatial dimensions: Piecewise smoothness of solutions and its breakdown. SIAM J. Numer. Anal., 17: 1178–1197, 1986.

    MathSciNet  MATH  Google Scholar 

  63. I. Lomtev and G.E. Karniadakis. A discontinuous Galerkin method for the Navier-Stokes equations. Int. J. Num. Meth. Fluids. in press.

    Google Scholar 

  64. I. Lomtev and G.E. Karniadakis. A discontinuous spectral/ hp element Galerkin method for the Navier-Stokes equations on unstructured grids. In Proc. IMACS WC’97, 1997. Berlin, Germany.

    Google Scholar 

  65. I. Lomtev and G.E. Karniadakis. Simulations of viscous supersonic flows on unstructured hp-meshes. AIAA-97–0754, 1997. 35th. Aerospace Sciences Meeting, Reno.

    Google Scholar 

  66. I. Lomtev, C.W. Quillen, and G.E. Karniadakis. Spectral/hp methods for viscous compressible flows on unstructured 2D meshes. J. Comput. Phys., 144: 325–357, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  67. E.O. Macagno and T. Hung. Computational and experimental study of a captive annular eddy.,J.F.M., 28: 43–XX, 1967.

    Google Scholar 

  68. X. Makridakis and I. Babusska. On the stability of the discontinuous Galerkin method for the heat equation. SIAM J. Numer. Anal., 34: 389–401, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  69. Newmann. A Computational Study of Fluid/Structure Interactions: Flow-Induced Vibrations of a Flexible Cable. PhD thesis, Princeton University, 1996.

    Google Scholar 

  70. J.T. Oden, No Babus’ka, and C.E. Baumann. A discontinuous hp finite element method for diffusion problems. J. Comput. Phys., 146: 491–519, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  71. S. Osher. Riemann solvers, the entropy condition and difference approximations. SIAM J. Numer. Anal., 21: 217–235, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  72. S. Osher and J. Sethian. Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulation. J. Comput. Phys., 79: 12–49, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  73. S. Osher and C.-W. Shu. High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal., 28: 907–922, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  74. C. Ozturan, H.L. deCougny, M.S. Shephard, and J.E. Flaherty. Parallel adaptive mesh refinement and redistribution on distributed memory computers. Comput. Methods Appl. Mech. Engrg., 119: 123–137, 1994.

    Article  Google Scholar 

  75. T. Peterson. A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal., 28: 133–140, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  76. W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73–479, Los Alamos Scientific Laboratory, 1973.

    Google Scholar 

  77. G.R. Richter. An optimal-order error estimate for the discontinuous Galerkin method. Math. Comp., 50: 75–88, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  78. E. Rouy and A. Tourin. A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal., 29: 867–884, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  79. S.J. Sherwin and G. Karniadakis. Thetrahedral hp-finite elements: Algorithms and flow simulations. J. Comput. Phys., 124: 314–345, 1996.

    Article  MathSciNet  Google Scholar 

  80. C.-W. Shu and S. Osher Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77: 439–471, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  81. C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock capturing schemes, II. J. Comput. Phys., 83: 32–78, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  82. C.W. Shu. TVB uniformly high order schemes for conservation laws. Math. Comp., 49: 105–121, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  83. C.W. Shu. TVD time discretizations. SIAM J. Sci. Stat. Comput., 9: 1073–1084, 1988.

    Article  MATH  Google Scholar 

  84. M. Sussman, P. Smereka, and S. Osher. A level set approach for computing solution to incompressible two-phase flow. J. Comput. Phys., 114: 146–159, 1994.

    Article  MATH  Google Scholar 

  85. C. Tong and G.Q. Chen. Some fundamental concepts about systems of two spatial dimensional conservation laws. Acta Mathematica Scientia (English Ed.), 6: 463–474, 1986.

    Google Scholar 

  86. C. Tong and Y.-X. Zheng. Two dimensional Riemann problems for a single conservation law. Trans. Amer. Math. Soc., 312: 589–619, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  87. J.R. Trujillo. Effective high-order vorticity-velocity formulation. PhD thesis, Princeton University, 1997.

    Google Scholar 

  88. B. van Leer. Towards the ultimate conservation difference scheme, II. J. Comput. Phys., 14: 361–376, 1974.

    Article  MATH  Google Scholar 

  89. B. van Leer. Towards the ultimate conservation difference scheme, V. J. Comput. Phys., 32: 1–136, 1979.

    Article  Google Scholar 

  90. D. Wagner. The Riemann problem in two space dimensions for a single conservation law. SIAM J. Math. Anal., 14: 534–559, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  91. T.C. Warburton, I. Lomtev, R.M. Kirby, and G.E. Karniadakis A discontinuous Galerkin method for the Navier-Stokes equations in hybrid grids. In M. Hafez and J.C. Heirich, editors, 10th. International Conference on Finite Elements in Fluids, Tucson, Arizona, 1998.

    Google Scholar 

  92. P. Woodward and P. Colella. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54: 115–173, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  93. A.-H. Zhou and Q. Lin. Optimal and superconvergence estimates of the finite element method for a scalar hyperbolic equation. Acta Math. Sci., 14: 90–94, 1994.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN, 55455, USA

    Bernardo Cockburn

Authors
  1. Bernardo Cockburn
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Information Sciences Directorate, NASA Ames Research Center, 94035, Moffett Field, CA, USA

    Timothy J. Barth

  2. Karman Institute for Fluid Dynamics, Waterloosesteenweg, 72, 1640, Sint-Genesius-Rode, Belgium

    Herman Deconinck

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Cockburn, B. (1999). Discontinuous Galerkin Methods for Convection-Dominated Problems. In: Barth, T.J., Deconinck, H. (eds) High-Order Methods for Computational Physics. Lecture Notes in Computational Science and Engineering, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03882-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-03882-6_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-03884-0

  • Online ISBN: 978-3-662-03882-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation