Journal of Scientific Computing

, Volume 16, Issue 3, pp 173–261

Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

  • Bernardo Cockburn
  • Chi-Wang Shu
Article

Abstract

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

discontinuous Galerkin methods non-linear conservation laws convection-diffusion equations 

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REFERENCES

  1. 1.
    R. Abgrall, (1996). Numerical discretization of the first-order Hamilton-Jacobi equations on triangular meshes. Comm.Pure Appl.Math. 49, 1339–1377.Google Scholar
  2. 2.
    Adjerid, S., Aiffa, M., and Flaherty, J. E. (1998). Computational methods for singularly perturbed systems. In Cronin, J., and O'Malley, R. E. (eds.), Singular Perturbation Concepts of Differential Equations, AMS Proceedings of Symposia in Applied Mathematics, AMS.Google Scholar
  3. 3.
    Adjerid, S., Aiffa, M., and Flaherty, J. E. (1995). High-order finite element methods for singularly-perturbed elliptic and parabolic problems. SIAM J.Appl.Math. 55, 520–543.Google Scholar
  4. 4.
    Adjerid, S., Flaherty, J. E., and Krivodonova, L. Superconvergence and a posteriori error estimation for continuous and discontinuous Galerkin methods applied to singularly perturbed parabolic and hyperbolic problems, in preparation.Google Scholar
  5. 5.
    Aizinger, V., Dawson, C. N., Cockburn, B., and Castillo, P. (2000). Local discontinuous Galerkin method for contaminant transport. Advances in Water Resources 24, 73–87.Google Scholar
  6. 6.
    Allmaras, S. R. (1989). A Coupled Euler/Navier-Stokes Algorithm for 2-D Unsteady Transonic Shock/Boundary-Layer Interaction, Ph.D. thesis, Massachussetts Institute of Technology.Google Scholar
  7. 7.
    Allmaras, S. R., and Giles, M. B. (1987). A Second Order Flux Split Scheme for the Unsteady 2-D Euler Equations on Arbitrary Meshes, 8th. AIAA Computational Fluid Dynamic Conference, Honolulu, Hawai, June 9–11. TIAIAA, 87–1119-CP.Google Scholar
  8. 8.
    Alotto, P., Bertoni, A., Perugia, I., and Schötzau, D. (2000). Discontinuous finite element methods for the simulation of rotating electrical machines, Proceedings of 9th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering, September 11–14, Graz, Austria.Google Scholar
  9. 9.
    Arnold, D. N. (1982). An interior penalty finite element method with discontinuous elements. SIAM J.Numer.Anal. 19, 742–760.Google Scholar
  10. 10.
    Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J.Numer.Anal., to appear.Google Scholar
  11. 11.
    Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, D. (2000). Discontinuous Galerkin methods for elliptic problems. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 89–101.Google Scholar
  12. 12.
    Atkins H. L., and Shu, C.-W. (1998). Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations. AIAA J. 36, 775–782.Google Scholar
  13. 13.
    Augoula, S., and Abgrall, R. (2000). High order numerical discretization for Hamilton-Jacobi equations on triangular meshes. J.Sci.Comput. 15, 197–229.Google Scholar
  14. 14.
    Babuška, I., and Zlámal, M. (1973). Nonconforming elements in the finite element method with penalty. SIAM J.Numer.Anal. 10, 863–875.Google Scholar
  15. 15.
    Baker, G. A. (1977). Finite element methods for elliptic equations using nonconforming elements. Math.Comp. 31, 45–59.Google Scholar
  16. 16.
    Baker, G. A., Jureidini, W. N., and Karakashian, O. A. (1990). Piecewise solenoidal vector fields and the Stokes problem. SIAM J.Numer.Anal. 27, 1466–1485.Google Scholar
  17. 17.
    Bardos, C., LeRoux, A. Y., and Nédélec, J. C. (1979). First order quasilinear equations with boundary conditions. Comm.in P.D.E.4, 1017–1034.Google Scholar
  18. 18.
    Bassi, F., and S. Rebay, (1997). A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J.Comput.Phys. 131, 267–279.Google Scholar
  19. 19.
    Bassi, F., and S. Rebay, (1997). High-order accurate discontinuous finite element solution of the 2DEuler equations. J.Comput.Phys. 138, 251–285.Google Scholar
  20. 20.
    Bassi, F., and S. Rebay, (2000). GMRES for discontinuous Galerkin solution of the compressible Navier-Stokes equations. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 197–208.Google Scholar
  21. 21.
    Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., and Savini, M. (1997). A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In Decuypere, R., and Dibelius, G. (eds.), 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics (Antwerpen, Belgium), March 5–7, Technologisch Instituut, pp. 99–108.Google Scholar
  22. 22.
    Baumann, C. E., and Oden, J. T. (1999). A discontinuous hpfinite element method for convection-diffusion problems. Comput.Methods Appl.Mech.Engrg. 175, 311–341.Google Scholar
  23. 23.
    Bernardi, C., Maday, Y., and Patera, A. T. (1993). Domain decomposition by the mortar element method. In Kaper, H. G., and Garbey, M. (eds.), Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, Kluwer Academic Publishers, pp. 269–286.Google Scholar
  24. 24.
    Bernardi, C., Maday, Y., and Patera, A. T. (1994). A new nonconforming approach to domain decomposition: The mortar element method. In Brézis, H., and Lions, J. L. (eds.), Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Volume XI, Pitman Research Notes in Mathematics, No. 299, Pitman Advanced Publishing Program.Google Scholar
  25. 25.
    Bernardi, C., Debit, N., and Maday, Y. (1990). Coupling finite element and spectral methods: First results. Math.Comp. 54, No. 189, 21–39.Google Scholar
  26. 26.
    Biswas, R., Devine, K. D., and Flaherty, J. (1994). Parallel, adaptive finite element methods for conservation laws. Appl.Numer.Math. 14, 255–283.Google Scholar
  27. 27.
    Bourgeat, A., and Cockburn, B. (1989). The TVD-projection method for solving implicit numerical schemes for scalar conservation laws: A numerical study of a simple case. SIAM J.Sci.Stat.Comput. 10, 253–273.Google Scholar
  28. 28.
    Bramble, J. H., and Schatz, A. H. (1977). Higher order local accuracy by averaging in the finite element method. Math.Comp. 31, 94–111.Google Scholar
  29. 29.
    Brezzi, F., Manzini, G., Marini, D., Pietra, P., and Russo, A., (2000). Discontinuous Galerkin approximations for elliptic problems. Numer.Methods Partial Differential Equations 16, 365–378.Google Scholar
  30. 30.
    Brezzi, F., Marini, D., Pietra, P., and Russo, A. (1999). Discontinuous finite elements for diffusion problems, Atti Convegno in onore di F. Brioschi (Milano 1997), Istituto Lombardo, Accademia di Scienze e Lettere, pp. 197–217.Google Scholar
  31. 31.
    Castillo, P. (2000). An optimal error estimate for the local discontinuous Galerkin method. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 285–290.Google Scholar
  32. 32.
    Castillo, P., Cockburn, B., Perugia, I., and Schötzau, D., (2000). An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J.Numer.Anal. 38, 1676–1706.Google Scholar
  33. 33.
    Castillo, P., Cockburn, B., Schötzau, D., and Schwab, C. An optimal a priori error estimate for the hp-version of the local discontinuous Galerkin method for convectiondiffusion problems. Math.Comp., to appear.Google Scholar
  34. 34.
    Chavent, G., and Cockburn, B. (1989). The local projection TIP 0 P 1-discontinuous-Galerkin finite element method for scalar conservation laws. RAIRO Modél.Math.Anal.Numér. 23, 565–592.Google Scholar
  35. 35.
    Chavent, G., and Salzano, G. (1982). A finite element method for the 1Dwater flooding problem with gravity. J.Comput.Phys. 45, 307–344.Google Scholar
  36. 36.
    Chen, Z., Cockburn, B., Gardner, C., and Jerome, J. (1995). Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode. J.Comput.Phys. 117, 274–280.Google Scholar
  37. 37.
    Chen, Z., Cockburn, B., Jerome, J., and Shu, C.-W. (1995). Mixed-RKDG finite element methods for the 2-Dhydrodynamic model for semiconductor device simulation. VLSI Design 3, 145–158.Google Scholar
  38. 38.
    Cockburn, B. (1999). Discontinuous Galerkin methods for convection-dominated problems. In Barth, T., and Deconink, H. (eds.), High-Order Methods for Computational Physics, Lecture Notes in Computational Science and Engineering, Vol. 9, Springer-Verlag, pp. 69–224.Google Scholar
  39. 39.
    Cockburn, B. (2001). Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. J.Comput.Appl.Math. 128, 187–204.Google Scholar
  40. 40.
    Cockburn, B., and Gremaud, P. A. (1996). Error estimates for finite element methods for nonlinear conservation laws. SIAM J.Numer.Anal. 33, 522–554.Google Scholar
  41. 41.
    Cockburn, B., Hou, S., and Shu, C.-W. (1990). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math.Comp. 54, 545–581.Google Scholar
  42. 42.
    Cockburn, B., Kanschat, G., Perugia, I., and Schötzau, D. (2001). Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J.Numer.Anal. 39, 264–285.Google Scholar
  43. 43.
    Cockburn, B., Kanschat, G., Schötzau, D., and Schwab, C. Local discontinuous Galerkin methods for the Stokes system. SIAM J.Numer.Anal., to appear.Google Scholar
  44. 44.
    Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (2000). The development of discontinuous Galerkin methods. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 3–50.Google Scholar
  45. 45.
    Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.) (2000). Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag.Google Scholar
  46. 46.
    Cockburn, B., Lin, S. Y., and Shu, C.-W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. J.Comput.Phys. 84, 90–113.Google Scholar
  47. 47.
    Cockburn, B., Luskin, M., Shu, C.-W., and Süli, E. Enhanced accuracy by post-processing for finite element methods for hyperbolic equations, Math.Comp., to appear.Google Scholar
  48. 48.
    Cockburn, B., and Shu, C.-W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework. Math.Comp. 52, 411–435.Google Scholar
  49. 49.
    Cockburn, B., and Shu, C.-W. (1991). The Runge-Kutta local projection TIP 1-discontinuous Galerkin method for scalar conservation laws. RAIRO Modél.Math.Anal.Numér. 25, 337–361.Google Scholar
  50. 50.
    Cockburn, B., and Shu, C.-W. (1998). The local discontinuous Galerkin method for timedependent convection-diffusion systems. SIAM J.Numer.Anal. 35, 2440–2463.Google Scholar
  51. 51.
    Cockburn, B., and Shu, C.-W. (1998). The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems. J.Comput.Phys. 141 (1998), 199–224.Google Scholar
  52. 52.
    Crandall, M., and Majda, A. (1980). Monotone difference approximations for scalar conservation laws. Math.Comp. 34, 1–21.Google Scholar
  53. 53.
    Crandall, M. G., and Lions, P. L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans.Amer.Math.Soc. 277, 1–42.Google Scholar
  54. 54.
    Dawson, C. N. (1995). High resolution upwind-mixed finite element methods for advection-diffusion equations with variable time-stepping. Numer.Methods Partial Differential Equations 11, 525–538.Google Scholar
  55. 55.
    Dawson, C. N., and Kirby, R. (2001). High resolution schemes for conservation laws with locally varying time steps. SIAM J.Math.Anal. 22, 2256–2281.Google Scholar
  56. 56.
    Douglas, Jr., J., Darlow, B. L., Kendall, R. P., and Wheeler, M. F. (1979). Self-adaptive Galerkin methods for one-dimensional, two-phase immiscible flow, AIME Fifth Simposium on Reservoir Simulation (Denver, Colorado), Society of Petroleum Engineers, pp. 65–72.Google Scholar
  57. 57.
    Douglas, Jr., J., and Dupont, T. (1976). Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, Lecture Notes in Physics, Vol. 58, Springer-Verlag, Berlin.Google Scholar
  58. 58.
    Dubiner, M. (1991). Spectral methods on triangles and other domains. J.Sci.Comp. 6, 345–390.Google Scholar
  59. 59.
    Falk, R. (2000). Analysis of finite element methods for linear hyperbolic problems. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 103–112.Google Scholar
  60. 60.
    Feng, X., and Karakashian, O. A. Two-level non-overlapping schwarz methods for a discontinuous Galerkin method. SIAM J.Numer.Anal., to appear.Google Scholar
  61. 61.
    Flaherty, J. E., Loy, R. M., Shephard, M. S., Szymanski, B. K., Teresco, J. D., and Ziantz, L. H. (1997). Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws. J.Parallel and Dist.Comput. 47, 139–152.Google Scholar
  62. 62.
    Gopalakrishnan, J., and Kanshat, G. A multilevel discontinuous Galerkin method. Numer.Math., to appear.Google Scholar
  63. 63.
    Gottlieb, S., and Shu, C.-W. (1998). Total variation diminishing Runge-Kutta schemes. Math.Comp. 67, 73–85.Google Scholar
  64. 64.
    Gottlieb, S., Shu, C.-W., and Tadmor, E. (2001). Strong stability preserving high order time discretization methods. SIAM Rev. 43, 89–112.Google Scholar
  65. 65.
    Halt, D. W. (1992). A Compact Higher Order Euler Solver for Unstructured Grids, Ph.D. thesis, Washington University.Google Scholar
  66. 66.
    Halt, D. W., and Agarwall, R. K. (1991). A compact higher order characteristic-based Euler solver for unstructured grids. AIAA, 91–3234.Google Scholar
  67. 67.
    Halt, D. W., and Agarwall, R. K. (1992). A compact higher order Euler solver for unstructured grids with curved boundaries. AIAA, 92–2696.Google Scholar
  68. 68.
    Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. J.Comput.Phys. 49, 357–393.Google Scholar
  69. 69.
    Harten, A., Hyman, J. M., and Lax, P. D. (1976). On finite difference approximations and entropy conditions for shocks. Comm.Pure and Appl.Math. 29, 297–322.Google Scholar
  70. 70.
    Houston, P., Schwab, C., and Süli, E. (2000). Stabilized hp-finite element methods for hyperbolic problems. SIAM J.Numer.Anal. 37, 1618–1643.Google Scholar
  71. 71.
    Hu, C., Lepsky, O., and Shu, C.-W. (2000). The effect of the lest square procedure for discontinuous Galerkin methods for Hamilton-Jacobi equations. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 343–348.Google Scholar
  72. 72.
    Hu, C., and Shu, C.-W. (1999). A discontinuous Galerkin finite element method for Hamilton-Jacobi equations. SIAM J.Sci.Comput. 21, 666–690.Google Scholar
  73. 73.
    Jaffré, J., Johnson, C., and Szepessy, A. (1995). Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math.Models Methods Appl.Sci. 5, 367–386.Google Scholar
  74. 74.
    Jiang, G., and Shu, C.-W. (1994). On a cell entropy inequality for discontinuous Galerkin methods. Math.Comp. 62, 531–538.Google Scholar
  75. 75.
    Jiang, G.-S., and Peng, D.-P. (2000). Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J.Sci.Comput. 21, 2126–2143.Google Scholar
  76. 76.
    Jin, S., and Xin, Z.-P. (1998). Numerical passage from systems of conservation laws to Hamilton-Jacobi equation. SIAM J.Numer.Anal. 35, 2385–2404.Google Scholar
  77. 77.
    Johnson, C., and Pitkäranta, J. (1986). An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math.Comp. 46, 1–26.Google Scholar
  78. 78.
    Karniadakis, G. E., and Sherwin, S. J. (1999). Spectral/hp Element Methods in CFD, Oxford University Press.Google Scholar
  79. 79.
    Kuznetsov, N. N. (1976). Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comp.Math.and Math.Phys. 16, 105–119.Google Scholar
  80. 80.
    Lafon, F., and Osher, S. (1996). High-order 2-dimensional nonoscillatory methods for solving Hamilton-Jacobi scalar equations. J.Comput.Phys. 123, 235–253.Google Scholar
  81. 81.
    Lasser, C., and Toselli, A. (2000). An Overlapping Domain Decomposition Preconditioner for a Class of Discontinuous Galerkin Approximations of Advection-Diffusion Problems, Tech. Report 2000–12, Seminar für Angewandte Mathematik, ETH Zürich.Google Scholar
  82. 82.
    Lepsky, O., Hu, C., and Shu, C.-W. (2000). Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations. Appl.Numer.Math. 33, 423–434.Google Scholar
  83. 83.
    LeSaint, P., and Raviart, P. A. (1974). On a finite element method for solving the neutron transport equation. In de Boor, C. (ed.), Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, pp. 89–145.Google Scholar
  84. 84.
    LeVeque, R. J. (1990). Numerical Methods for Conservation Laws, Birkhäuser.Google Scholar
  85. 85.
    Lin, Q. (2000). Full convergence for hyperbolic finite elements. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 167–177.Google Scholar
  86. 86.
    Lin, Q., Yan, N., and Zhou, A.-H. (1996). An optimal error estimate of the discontinuous Galerkin method. J.Engrg.Math. 13, 101–105.Google Scholar
  87. 87.
    Lin, Q., and Zhou, A.-H. (1993). Convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. Acta Math.Sci. 13, 207–210.Google Scholar
  88. 88.
    Liu, J.-G., and Shu, C.-W. (2000). A high order discontinuous Galerkin method for 2D incompressible flows. J.Comput.Phys. 160, 577–596.Google Scholar
  89. 89.
    Liu, J.-G., and Shu, C.-W. (2000). A numerical example on the performance of highorder discontinuous Galerkin method for 2Dincompressible flows. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 369–374.Google Scholar
  90. 90.
    Liu, J.-G., and Xin, Z.-P. (2000). Convergence of a Galerkin method for 2Ddiscontinuous Euler flows. Comm.Pure Appl.Math. 53, 786–798.Google Scholar
  91. 91.
    Lomtev, I., Kirby, R. M., and Karniadakis, G. E. (2000). A discontinuous Galerkin method in moving domains. In Cockburn, B., Karniadakis, G. E., and Shu, C.-W. (eds.), Discontinuous Galerkin Methods.Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, pp. 375–383.Google Scholar
  92. 92.
    Oden, J. T., Babuška, I., and Baumann, C. E. (1998). A discontinuous hpfinite element method for diffusion problems. J.Comput.Phys. 146, 491–519.Google Scholar
  93. 93.
    Osher, S. (1984). Convergence of generalized MUSCL schemes. SIAM J.Numer.Anal. 22, 947–961.Google Scholar
  94. 94.
    Osher, S., and Sethian, J. A. (1988). Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J.Comput.Phys. 79, 12–49.Google Scholar
  95. 95.
    Osher, S., and Shu, C.-W. (1991). High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J.Numer.Anal. 28, 907–922.Google Scholar
  96. 96.
    Perugia, I. and Schötzau, D. The coupling of local discontinuous Galerkin and conforming finite element methods. J.Sci.Comput., to appear.Google Scholar
  97. 97.
    Peterson, T. (1991). A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J.Numer.Anal. 28, 133–140.Google Scholar
  98. 98.
    Reed, W. H., and Hill, T. R. Triangular Mesh Methods for the Neutron Transport Equation, Tech. Report LA-UR–73–479, Los Alamos Scientific Laboratory, 1973.Google Scholar
  99. 99.
    Richter, G. R. (1988). An optimal-order error estimate for the discontinuous Galerkin method. Math.Comp. 50, 75–88.Google Scholar
  100. 100.
    Rivière, B., Wheeler, M. F., and Girault, V. (1999). Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Comput.Geom. 3, 337–360.Google Scholar
  101. 101.
    Rouy, E., and Tourin, A. (1992). A viscosity solutions approach to shape-from-shading. SIAM J.Numer.Anal. 29, 867–884.Google Scholar
  102. 102.
    Sanders, R. (1983). On convergence of monotone finite difference schemes with variable spacing differencing. Math.Comp. 40, 91–106.Google Scholar
  103. 103.
    Schwab, C. (1999). hp-FEM for fluid flow simulation. In Barth, T., and Deconink, H. (eds.), High-Order Methods for Computational Physics, Lecture Notes in Computational Science and Engineering, Vol. 9, Springer-Verlag, pp. 325–438.Google Scholar
  104. 104.
    Shu, C.-W. (1987). TVB boundary treatment for numerical solutions of conservation laws. Math.Comp. 49, 123–134.Google Scholar
  105. 105.
    Shu, C.-W. (1987). TVB uniformly high order schemes for conservation laws. Math.Comp. 49, 105–121.Google Scholar
  106. 106.
    Shu, C.-W. (1987). TVDtime discretizations. SIAM J.Sci.Stat.Comput. 9 (1988), 1073–1084.Google Scholar
  107. 107.
    Shu, C.-W. (2001). Different formulations of the discontinuous Galerkin method for the viscous terms. In Shi, Z.-C., Mu, M., Xue, W., and Zou, J. (eds.), Advances in Scientific Computing, Science Press, pp. 144–155.Google Scholar
  108. 108.
    Shu, C.-W., and Osher, S. (1988). Efficient implementation of essentially non-oscillatory shock-capturing schemes. J.Comput.Phys. 77, 439–471.Google Scholar
  109. 109.
    Shu, C.-W., and Osher, S. (1989). Efficient implementation of essentially non-oscillatory shock capturing schemes, II. J.Comput.Phys. 83, 32–78.Google Scholar
  110. 110.
    Strang, G., and Fix, G. (1973). An Analysis of the Finite Element Method, Prentice-Hall, New Jersey.Google Scholar
  111. 111.
    van Leer, B. (1974). Towards the ultimate conservation difference scheme, II. J.Comput.Phys. 14, 361–376.Google Scholar
  112. 112.
    van Leer, B. (1979). Towards the ultimate conservation difference scheme, V. J.Comput.Phys. 32, 1–136.Google Scholar
  113. 113.
    Warburton, T. C. (1998). Spectral/hp Methods on Polymorphic Multi-Domains: Algorithms and Applications, Ph.D. thesis, Brown University.Google Scholar
  114. 114.
    Wheeler, M. F. (1978). An elliptic collocation-finite element method with interior penalties. SIAM J.Numer.Anal. 15, 152–161.Google Scholar
  115. 115.
    Wierse, M. (1997). A new theoretically motivated higher order upwind scheme on unstructured grids of simplices. Adv.Comput.Math. 7, 303–335.Google Scholar
  116. 116.
    Woodward, P., and Colella, P. (1984). The numerical simulation of two-dimensional fluid flow with strong shocks. J.Comput.Phys. 54, 115–173.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Bernardo Cockburn
    • 1
  • Chi-Wang Shu
    • 2
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolis
  2. 2.Division of Applied MathematicsBrown UniversityProvidence

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