Advertisement

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

  • Chi-Wang Shu
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (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.

Keywords

Discontinuous Galerkin method Time-dependent partial differential equations Superconvergence Positivity-preserving δ-functions. 

Notes

7 Acknowledgement

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

References

  1. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Adjerid and T. Weinhart, Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems, Mathematics of Computations, 80 (2011), pp. 1335–1367.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 39 (1982), pp. 742–760.CrossRefGoogle Scholar
  7. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [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/s00211-013-0555-3Google Scholar
  12. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [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.CrossRefGoogle Scholar
  15. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [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.CrossRefGoogle Scholar
  20. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [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. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [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.MathSciNetzbMATHGoogle Scholar
  24. [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. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [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.MathSciNetzbMATHGoogle Scholar
  28. [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. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [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.MathSciNetGoogle Scholar
  33. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    C. Dawson, Foreword for the special issue on discontinuous Galerkin method, Computer Methods in Applied Mechanics and Engineering, 195 (2006), p. 3183.CrossRefGoogle Scholar
  36. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Harten, High resolution schemes for hyperbolic conservation laws, Journal of Computational Physics, 49 (1983), pp. 357–393.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods, Springer, New York, 2008.CrossRefzbMATHGoogle Scholar
  39. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    C. Hu and C.-W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, Journal of Computational Physics, 150 (1999), pp. 97–127.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [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.CrossRefGoogle Scholar
  45. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [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.MathSciNetzbMATHGoogle Scholar
  48. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    G.-S. Jiang and C.-W. Shu, On cell entropy inequality for discontinuous Galerkin methods, Mathematics of Computation, 62 (1994), pp. 531–538.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126 (1996), pp. 202–228.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    G. Kanschat, Discontinuous Galerkin Methods for Viscous Flow, Deutscher Universitätsverlag, Wiesbaden, 2007.Google Scholar
  55. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [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. [57]
    R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel, 1990.CrossRefzbMATHGoogle Scholar
  58. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    B. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Birkhauser, Basel, 2006.zbMATHGoogle Scholar
  60. [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.CrossRefzbMATHGoogle Scholar
  61. [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.MathSciNetGoogle Scholar
  62. [62]
    X. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115 (1994), pp. 200–212.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  64. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  66. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  68. [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.MathSciNetCrossRefGoogle Scholar
  69. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  70. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  71. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  72. [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. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    G.R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Mathematics of Computation, 50 (1988), pp. 75–88.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    B. Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations. Theory and implementation, SIAM, Philadelphia, 2008.Google Scholar
  76. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  77. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  78. [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.CrossRefzbMATHGoogle Scholar
  79. [79]
    C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Mathematics of Computation, 49 (1987), pp. 105–121.MathSciNetCrossRefzbMATHGoogle Scholar
  80. [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. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  82. [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. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  84. [84]
    M.F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), pp. 152–161.MathSciNetCrossRefzbMATHGoogle Scholar
  85. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  86. [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.MathSciNetGoogle Scholar
  87. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  88. [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.CrossRefGoogle Scholar
  89. [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.MathSciNetzbMATHGoogle Scholar
  90. [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.MathSciNetzbMATHGoogle Scholar
  91. [91]
    Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, Journal of Computational Physics, 205 (2005), pp. 72–97.MathSciNetCrossRefzbMATHGoogle Scholar
  92. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  93. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  94. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  95. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  96. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  97. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  98. [98]
    Y. Xu and C.-W. Shu, Dissipative numerical methods for the Hunter-Saxton equation, Journal of Computational Mathematics, 28 (2010), pp. 606–620.MathSciNetCrossRefzbMATHGoogle Scholar
  99. [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.MathSciNetGoogle Scholar
  100. [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.MathSciNetGoogle Scholar
  101. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  102. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  103. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  104. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  105. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  106. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  107. [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. [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.MathSciNetCrossRefGoogle Scholar
  109. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  110. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  111. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  112. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  113. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  114. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  115. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  116. [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.MathSciNetGoogle Scholar
  117. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  118. [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.MathSciNetCrossRefGoogle Scholar
  119. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  120. [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.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations