Skip to main content
SpringerLink
Log in

Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments

  • Chapter
  • First Online:
Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 157))

Abstract

In these lectures we give a general survey on discontinuous Galerkin methods for solving time-dependent partial differential equations. We also present a few recent developments on the design, analysis, and application of these discontinuous Galerkin methods.

AMS(MOS) subject classifications. Primary 65M60, 65M20, 65M12, 65M15.

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

Access this chapter

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 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 59.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Adjerid and M. Baccouch, Asymptotically exact a posteriori error estimates for a one-dimensional linear hyperbolic problem, Applied Numerical Mathematics, 60 (2010), pp. 903–914.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Adjerid, K. Devine, J. Flaherty and L. Krivodonova, A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Computational Methods in Applied Mechanics and Engineering, 191 (2002), pp. 1097–1112.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Adjerid and T. Massey, Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Computer Methods in Applied Mechanics and Engineering, 195 (2006), pp. 3331–3346.

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Adjerid and T. Weinhart, Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 3113–3129.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Adjerid and T. Weinhart, Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems, Mathematics of Computations, 80 (2011), pp. 1335–1367.

    Article  MathSciNet  MATH  Google Scholar 

  6. D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 39 (1982), pp. 742–760.

    Article  Google Scholar 

  7. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, Journal of Computational Physics, 131 (1997), pp. 267–279.

    Article  MathSciNet  MATH  Google Scholar 

  8. C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 175 (1999), pp. 311–341.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Biswas, K.D. Devine and J. Flaherty, Parallel, adaptive finite element methods for conservation laws, Applied Numerical Mathematics, 14 (1994), pp. 255–283.

    Article  MathSciNet  MATH  Google Scholar 

  10. O. Bokanowski, Y. Cheng and C.-W. Shu, A discontinuous Galerkin solver for front propagation, SIAM Journal on Scientific Computing, 33 (2011), pp. 923–938.

    Article  MathSciNet  MATH  Google Scholar 

  11. O. Bokanowski, Y. Cheng and C.-W. Shu, A discontinuous Galerkin scheme for front propagation with obstacles, Numerische Mathematik, to appear. DOI: 10.1007/s00211-013-0555-3

    Google Scholar 

  12. J.H. Bramble and A.H. Schatz, High order local accuracy by averaging in the finite element method, Mathematics of Computation, 31 (1977), pp. 94–111.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Burbeau, P. Sagaut and Ch.H. Bruneau, A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods, Journal of Computational Physics, 169 (2001), pp. 111–150.

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Cheng, I.M. Gamba, A. Majorana and C.-W. Shu, Discontinuous Galerkin solver for Boltzmann-Poisson transients, Journal of Computational Electronics, 7 (2008), pp. 119–123.

    Article  Google Scholar 

  15. Y. Cheng, I.M. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for Boltzmann Poisson systems in nano devices, Computer Methods in Applied Mechanics and Engineering, 198 (2009), pp. 3130–3150.

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, Journal of Computational Physics, 223 (2007), pp. 398–415.

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Mathematics of Computation, 77 (2008), pp. 699–730.

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Cheng and C.-W. Shu, Superconvergence and time evolution of discontinuous Galerkin finite element solutions, Journal of Computational Physics, 227 (2008), pp. 9612–9627.

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. Cheng and C.-W. Shu, Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-diffusion equations, Computers & Structures, 87 (2009), pp. 630–641.

    Article  Google Scholar 

  20. Y. Cheng and C.-W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), pp. 4044–4072.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, in High-Order Methods for Computational Physics, T.J. Barth and H. Deconinck (eds.), Lecture Notes in Computational Science and Engineering, volume 9, Springer, 1999, pp. 69–224.

    Google Scholar 

  22. B. Cockburn, B. Dong and J. Guzmán, Optimal convergence of the original DG method for the transport-reaction equation on special meshes, SIAM Journal on Numerical Analysis, 46 (2008), pp. 1250–1265.

    Article  MathSciNet  MATH  Google Scholar 

  23. 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, Mathematics of Computation, 54 (1990), pp. 545–581.

    MathSciNet  MATH  Google Scholar 

  24. B. Cockburn, G. Karniadakis and C.-W. Shu, The development of discontinuous Galerkin methods, in Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G. Karniadakis and C.-W. Shu (eds.), Lecture Notes in Computational Science and Engineering, volume 11, Springer, 2000, Part I: Overview, pp. 3–50.

    Google Scholar 

  25. 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, Journal of Computational Physics, 84 (1989), pp. 90–113.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Cockburn, M. Luskin, C.-W. Shu and E. Süli, Enhanced accuracy by post-processing for finite element methods for hyperbolic equations, Mathematics of Computation, 72 (2003), pp. 577–606.

    Article  MathSciNet  MATH  Google Scholar 

  27. B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of Computation, 52 (1989), pp. 411–435.

    MathSciNet  MATH  Google Scholar 

  28. B. Cockburn and C.-W. Shu, The Runge-Kutta local projection P 1 -discontinuous-Galerkin finite element method for scalar conservation laws, Mathematical Modelling and Numerical Analysis (M 2 AN), 25 (1991), pp. 337–361.

    Google Scholar 

  29. B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of Computational Physics, 141 (1998), pp. 199–224.

    Article  MathSciNet  MATH  Google Scholar 

  30. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection diffusion systems, SIAM Journal on Numerical Analysis, 35 (1998), pp. 2440–2463.

    Article  MathSciNet  MATH  Google Scholar 

  31. B. Cockburn and C.-W. Shu, Runge-Kutta Discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing, 16 (2001), pp. 173–261.

    Article  MathSciNet  MATH  Google Scholar 

  32. B. Cockburn and C.-W. Shu, Foreword for the special issue on discontinuous Galerkin method, Journal of Scientific Computing, 22–23 (2005), pp. 1–3.

    MathSciNet  Google Scholar 

  33. B. Cockburn and C.-W. Shu, Foreword for the special issue on discontinuous Galerkin method, Journal of Scientific Computing, 40 (2009), pp. 1–3.

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Curtis, R.M. Kirby, J.K. Ryan and C.-W. Shu, Post-processing for the discontinuous Galerkin method over non-uniform meshes, SIAM Journal on Scientific Computing, 30 (2007), pp. 272–289.

    Article  MathSciNet  MATH  Google Scholar 

  35. C. Dawson, Foreword for the special issue on discontinuous Galerkin method, Computer Methods in Applied Mechanics and Engineering, 195 (2006), p. 3183.

    Article  Google Scholar 

  36. B. Dong and C.-W. Shu, Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems, SIAM Journal on Numerical Analysis, 47 (2009), pp. 3240–3268.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Harten, High resolution schemes for hyperbolic conservation laws, Journal of Computational Physics, 49 (1983), pp. 357–393.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods, Springer, New York, 2008.

    Book  MATH  Google Scholar 

  39. S. Hou and X.-D. Liu, Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method, Journal of Scientific Computing, 31 (2007), pp. 127–151.

    Article  MathSciNet  MATH  Google Scholar 

  40. C. Hu and C.-W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, Journal of Computational Physics, 150 (1999), pp. 97–127.

    Article  MathSciNet  MATH  Google Scholar 

  41. C. Hu and C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM Journal on Scientific Computing, 21 (1999), pp. 666–690.

    Article  MathSciNet  MATH  Google Scholar 

  42. O. Lepsky, C. Hu and C.-W. Shu, Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations, Applied Numerical Mathematics, 33 (2000), pp. 423–434.

    Article  MathSciNet  MATH  Google Scholar 

  43. F. Li and C.-W. Shu, Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations, Applied Mathematics Letters, 18 (2005), pp. 1204–1209.

    Article  MathSciNet  MATH  Google Scholar 

  44. Y.-X. Liu and C.-W. Shu, Local discontinuous Galerkin methods for moment models in device simulations: formulation and one dimensional results, Journal of Computational Electronics, 3 (2004), pp. 263–267.

    Article  Google Scholar 

  45. Y.-X. Liu and C.-W. Shu, Local discontinuous Galerkin methods for moment models in device simulations: Performance assessment and two dimensional results, Applied Numerical Mathematics, 57 (2007), pp. 629–645.

    Article  MathSciNet  MATH  Google Scholar 

  46. Y.-X. Liu and C.-W. Shu, Error analysis of the semi-discrete local discontinuous Galerkin method for semiconductor device simulation models, Science China Mathematics, 53 (2010), pp. 3255–3278.

    Article  MathSciNet  MATH  Google Scholar 

  47. L. Ji and Y. Xu, Optimal error estimates of the local discontinuous Galerkin method for Willmore flow of graphs on Cartesian meshes, International Journal of Numerical Analysis and Modeling, 8 (2011), pp. 252–283.

    MathSciNet  MATH  Google Scholar 

  48. L. Ji and Y. Xu, Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on Cartesian meshes, Journal of Scientific Computing, 51 (2012), pp. 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  49. L. Ji, Y. Xu and J. Ryan, Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions, Mathematics of Computation, 81 (2012), pp. 1929–1950.

    Article  MathSciNet  MATH  Google Scholar 

  50. L. Ji, Y. Xu and J. Ryan, Negative order norm estimates for nonlinear hyperbolic conservation laws, Journal of Scientific Computing, 54 (2013), pp. 531–548.

    Article  MathSciNet  MATH  Google Scholar 

  51. G.-S. Jiang and C.-W. Shu, On cell entropy inequality for discontinuous Galerkin methods, Mathematics of Computation, 62 (1994), pp. 531–538.

    Article  MathSciNet  MATH  Google Scholar 

  52. G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126 (1996), pp. 202–228.

    Article  MathSciNet  MATH  Google Scholar 

  53. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Mathematics of Computation, 46 (1986), pp. 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  54. G. Kanschat, Discontinuous Galerkin Methods for Viscous Flow, Deutscher Universitätsverlag, Wiesbaden, 2007.

    Google Scholar 

  55. L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon and J.E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Applied Numerical Mathematics, 48 (2004), pp. 323–338.

    Article  MathSciNet  MATH  Google Scholar 

  56. P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical aspects of finite elements in partial differential equations, C. de Boor (ed.), Academic Press, 1974, pp. 89–145.

    Google Scholar 

  57. R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel, 1990.

    Book  MATH  Google Scholar 

  58. D. Levy, C.-W. Shu and J. Yan, Local discontinuous Galerkin methods for nonlinear dispersive equations, Journal of Computational Physics, 196 (2004), pp. 751–772.

    Article  MathSciNet  MATH  Google Scholar 

  59. B. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Birkhauser, Basel, 2006.

    MATH  Google Scholar 

  60. H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM Journal on Numerical Analysis, 47 (2009), pp. 675–698.

    Article  MATH  Google Scholar 

  61. H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion with interface corrections, Communications in Computational Physics, 8 (2010), pp. 541–564.

    MathSciNet  Google Scholar 

  62. X. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115 (1994), pp. 200–212.

    Article  MathSciNet  MATH  Google Scholar 

  63. X. Meng, C.-W. Shu and B. Wu, Superconvergence of the local discontinuous Galerkin method for linear fourth order time dependent problems in one space dimension, IMA Journal of Numerical Analysis, 32 (2012), pp. 1294–1328.

    Article  MathSciNet  MATH  Google Scholar 

  64. X. Meng, C.-W. Shu, Q. Zhang and B. Wu, Superconvergence of discontinuous Galerkin method for scalar nonlinear conservation laws in one space dimension, SIAM Journal on Numerical Analysis, 50 (2012), pp. 2336–2356.

    Article  MathSciNet  MATH  Google Scholar 

  65. H. Mirzaee, L. Ji, J. Ryan and R.M. Kirby, Smoothness-increasing accuracy-conserving (SIAC) post-processing for discontinuous Galerkin solutions over structured triangular meshes, SIAM Journal on Numerical Analysis, 49 (2011), pp. 1899–1920.

    Article  MathSciNet  MATH  Google Scholar 

  66. J.T. Oden, I. Babuvska and C.E. Baumann, A discontinuous hp finite element method for diffusion problems, Journal of Computational Physics, 146 (1998), pp. 491–519.

    Article  MathSciNet  MATH  Google Scholar 

  67. T. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM Journal on Numerical Analysis, 28 (1991), pp. 133–140.

    Article  MathSciNet  MATH  Google Scholar 

  68. J. Qiu and C.-W. Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one dimensional case, Journal of Computational Physics, 193 (2003), pp. 115–135.

    Article  MathSciNet  Google Scholar 

  69. J. Qiu and C.-W. Shu, A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters, SIAM Journal on Scientific Computing, 27 (2005), pp. 995–1013.

    Article  MathSciNet  MATH  Google Scholar 

  70. J. Qiu and C.-W. Shu, Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing, 26 (2005), pp. 907–929.

    Article  MathSciNet  MATH  Google Scholar 

  71. J. Qiu and C.-W. Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: two dimensional case, Computers & Fluids, 34 (2005), pp. 642–663.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  73. J.-F. Remacle, J. Flaherty and M. Shephard, An adaptive discontinuous Galerkin technique with an orthogonal basis applied to Rayleigh-Taylor flow instabilities, SIAM Review, 45 (2003), pp. 53–72.

    Article  MathSciNet  MATH  Google Scholar 

  74. G.R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Mathematics of Computation, 50 (1988), pp. 75–88.

    Article  MathSciNet  MATH  Google Scholar 

  75. B. Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations. Theory and implementation, SIAM, Philadelphia, 2008.

    Google Scholar 

  76. J. Ryan and C.-W. Shu, On a one-sided post-processing technique for the discontinuous Galerkin methods, Methods and Applications of Analysis, 10 (2003), pp. 295–308.

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Ryan, C.-W. Shu and H. Atkins, Extension of a postprocessing technique for the discontinuous Galerkin method for hyperbolic equations with application to an aeroacoustic problem, SIAM Journal on Scientific Computing, 26 (2005), pp. 821–843.

    Article  MathSciNet  MATH  Google Scholar 

  78. J. Shi, C. Hu and C.-W. Shu, A technique of treating negative weights in WENO schemes, Journal of Computational Physics, 175 (2002), pp. 108–127.

    Article  MATH  Google Scholar 

  79. C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Mathematics of Computation, 49 (1987), pp. 105–121.

    Article  MathSciNet  MATH  Google Scholar 

  80. C.-W. Shu, Discontinuous Galerkin methods: general approach and stability, Numerical Solutions of Partial Differential Equations, S. Bertoluzza, S. Falletta, G. Russo and C.-W. Shu, Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, Basel, 2009, pp. 149–201.

    Google Scholar 

  81. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), pp. 439–471.

    Article  MathSciNet  MATH  Google Scholar 

  82. M. Steffan, S. Curtis, R.M. Kirby and J. Ryan, Investigation of smoothness enhancing accuracy-conserving filters for improving streamline integration through discontinuous fields, IEEE-TVCG, 14 (2008), pp. 680–692.

    Google Scholar 

  83. C. Wang, X. Zhang, C.-W. Shu and J. Ning, Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations, Journal of Computational Physics, 231 (2012), pp. 653–665.

    Article  MathSciNet  MATH  Google Scholar 

  84. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), pp. 152–161.

    Article  MathSciNet  MATH  Google Scholar 

  85. Y. Xia, Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, Journal of Computational Physics, 227 (2007), pp. 472–491.

    Article  MathSciNet  MATH  Google Scholar 

  86. Y. Xia, Y. Xu and C.-W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Communications in Computational Physics, 5 (2009), pp. 821–835.

    MathSciNet  Google Scholar 

  87. Y. Xia, Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for the generalized Zakharov system, Journal of Computational Physics, 229 (2010), pp. 1238–1259.

    Article  MathSciNet  MATH  Google Scholar 

  88. Y. Xing, X. Zhang and C.-W. Shu, Positivity preserving high order well balanced discontinuous Galerkin methods for the shallow water equations, Advances in Water Resources, 33 (2010), pp. 1476–1493.

    Article  Google Scholar 

  89. T. Xiong, C.-W. Shu and M. Zhang, A priori error estimates for semi-discrete discontinuous Galerkin methods solving nonlinear Hamilton-Jacobi equations with smooth solutions, International Journal of Numerical Analysis and Modeling, 10 (2013), pp. 154–177.

    MathSciNet  MATH  Google Scholar 

  90. Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for three classes of nonlinear wave equations, Journal of Computational Mathematics, 22 (2004), pp. 250–274.

    MathSciNet  MATH  Google Scholar 

  91. Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, Journal of Computational Physics, 205 (2005), pp. 72–97.

    Article  MathSciNet  MATH  Google Scholar 

  92. Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for two classes of two dimensional nonlinear wave equations, Physica D, 208 (2005), pp. 21–58.

    Article  MathSciNet  MATH  Google Scholar 

  93. Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations, Computer Methods in Applied Mechanics and Engineering, 195 (2006), pp. 3430–3447.

    Article  MathSciNet  MATH  Google Scholar 

  94. Y. Xu and C.-W. Shu, Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Computer Methods in Applied Mechanics and Engineering, 196 (2007), pp. 3805–3822.

    Article  MathSciNet  MATH  Google Scholar 

  95. Y. Xu and C.-W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM Journal on Numerical Analysis, 46 (2008), pp. 1998–2021.

    Article  MathSciNet  MATH  Google Scholar 

  96. Y. Xu and C.-W. Shu, Local discontinuous Galerkin method for the Hunter-Saxton equation and its zero-viscosity and zero-dispersion limit, SIAM Journal on Scientific Computing, 31 (2008), pp. 1249–1268.

    Article  MathSciNet  MATH  Google Scholar 

  97. Y. Xu and C.-W. Shu, Local discontinuous Galerkin method for surface diffusion and Willmore flow of graphs, Journal of Scientific Computing, 40 (2009), pp. 375–390.

    Article  MathSciNet  MATH  Google Scholar 

  98. Y. Xu and C.-W. Shu, Dissipative numerical methods for the Hunter-Saxton equation, Journal of Computational Mathematics, 28 (2010), pp. 606–620.

    Article  MathSciNet  MATH  Google Scholar 

  99. Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Communications in Computational Physics, 7 (2010), pp. 1–46.

    MathSciNet  Google Scholar 

  100. Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for the Degasperis-Procesi equation, Communications in Computational Physics, 10 (2011), pp. 474–508.

    MathSciNet  Google Scholar 

  101. Y. Xu and C.-W. Shu, Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations, SIAM Journal on Numerical Analysis, 50 (2012), pp. 79–104.

    Article  MathSciNet  MATH  Google Scholar 

  102. J. Yan and S. Osher, A local discontinuous Galerkin method for directly solving HamiltonJacobi equations, Journal of Computational Physics, 230 (2011), pp. 232–244.

    Article  MathSciNet  MATH  Google Scholar 

  103. J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis, 40 (2002), pp. 769–791.

    Article  MathSciNet  MATH  Google Scholar 

  104. J. Yan and C.-W. Shu, Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, Journal of Scientific Computing, 17 (2002), pp. 27–47.

    Article  MathSciNet  MATH  Google Scholar 

  105. Y. Yang and C.-W. Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM Journal on Numerical Analysis, 50 (2012), pp. 3110–3133.

    Article  MathSciNet  MATH  Google Scholar 

  106. Y. Yang and C.-W. Shu, Discontinuous Galerkin method for hyperbolic equations involving δ-singularities: negative-order norm error estimates and applications, Numerische Mathematik, 124 (2013), pp. 753–781.

    Article  MathSciNet  MATH  Google Scholar 

  107. M. Zhang and C.-W. Shu, An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations, Mathematical Models and Methods in Applied Sciences (M 3 AS), 13 (2003), pp. 395–413.

    Google Scholar 

  108. Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM Journal on Numerical Analysis, 42 (2004), pp. 641–666.

    Article  MathSciNet  Google Scholar 

  109. Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws, SIAM Journal on Numerical Analysis, 44 (2006), pp. 1703–1720.

    Article  MathSciNet  MATH  Google Scholar 

  110. Q. Zhang and C.-W. Shu, Stability analysis and a priori error estimates to the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM Journal on Numerical Analysis, 48 (2010), pp. 1038–1063.

    Article  MathSciNet  MATH  Google Scholar 

  111. X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, Journal of Computational Physics, 229 (2010), pp. 3091–3120.

    Article  MathSciNet  MATH  Google Scholar 

  112. X. Zhang and C.-W. Shu, On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, Journal of Computational Physics, 229 (2010), pp. 8918–8934.

    Article  MathSciNet  MATH  Google Scholar 

  113. X. Zhang and C.-W. Shu, Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms, Journal of Computational Physics, 230 (2011), pp. 1238–1248.

    Article  MathSciNet  MATH  Google Scholar 

  114. X. Zhang and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order schemes for conservation laws: Survey and new developments, Proceedings of the Royal Society A, 467 (2011), pp. 2752–2776.

    Article  MathSciNet  MATH  Google Scholar 

  115. X. Zhang, Y. Xia and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, Journal of Scientific Computing, 50 (2012), pp. 29–62.

    Article  MathSciNet  MATH  Google Scholar 

  116. Y.-T. Zhang and C.-W. Shu, Third order WENO scheme on three dimensional tetrahedral meshes, Communications in Computational Physics, 5 (2009), pp. 836–848.

    MathSciNet  Google Scholar 

  117. X. Zhong and C.-W. Shu, Numerical resolution of discontinuous Galerkin methods for time dependent wave equations, Computer Methods in Applied Mechanics and Engineering, 200 (2011), pp. 2814–2827.

    Article  MathSciNet  MATH  Google Scholar 

  118. X. Zhong and C.-W. Shu, A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods, Journal of Computational Physics, 232 (2012), pp. 397–415.

    Article  MathSciNet  Google Scholar 

  119. J. Zhu, J.-X. Qiu, C.-W. Shu and M. Dumbser, Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes, Journal of Computational Physics, 227 (2008), pp. 4330–4353.

    Article  MathSciNet  MATH  Google Scholar 

  120. J. Zhu, X. Zhong, C.-W. Shu and J.-X. Qiu, Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured mesh, Journal of Computational Physics, Numerische Mathematik, 124 (2013), pp. 753–781.

    Article  Google Scholar 

Download references

7 Acknowledgement

The work of the author was supported in part by NSF grant DMS-1112700 and DOE grant DE-FG02-08ER25863.

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Chi-Wang Shu

Authors
  1. Chi-Wang Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chi-Wang Shu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Xiaobing Feng

  2. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Ohannes Karakashian

  3. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Yulong Xing

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Shu, CW. (2014). Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments. In: Feng, X., Karakashian, O., Xing, Y. (eds) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-01818-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation