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All issues Volume 49 / No 4 (July-August 2015) ESAIM: M2AN, 49 4 (2015) 991-1018 References
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Issue
ESAIM: M2AN
Volume 49, Number 4, July-August 2015
Page(s) 991 - 1018
DOI https://doi.org/10.1051/m2an/2014063
Published online 19 June 2015
  1. R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). [Google Scholar]
  2. E. Burman, A. Ern and M.A. Fernandez, Explicit Runge–Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems. SIAM. J. Numer. Anal. 48 (2010) 2019–2042. [Google Scholar]
  3. P.G. Ciarlet, Finite Element Method for Elliptic Problems. North–Holland, Amsterdam (1978). [Google Scholar]
  4. B. Cockburn and J. Guzmán, Error estimates for the Runge–Kutta discontinuous Galerkin method for the transport equation with discontinuous initial data. SIAM. J. Numer. Anal. 46 (2008) 1364–1398. [CrossRef] [MathSciNet] [Google Scholar]
  5. B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework. Math. Comput. 52 (1989) 411–435. [Google Scholar]
  6. B. Cockburn and C.-W. Shu, The Runge–Kutta local projection P1-discontinuous Galerkin method for scalar conservation laws. RAIRO Modél. Math. Anal. Numér. 25 (1991) 337–361. [Google Scholar]
  7. B. Cockburn and C.-W. Shu, The Runge–Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems J. Comput. Phys. 141 (1998a) 199–224. [CrossRef] [MathSciNet] [Google Scholar]
  8. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems SIAM. J. Numer. Anal. 35 (1998b) 2440–2463. [Google Scholar]
  9. B. Cockburn and C.-W. Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems J. Sci. Comput. 16 (2001) 173–261. [CrossRef] [MathSciNet] [Google Scholar]
  10. 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 (1990) 545–581. [MathSciNet] [Google Scholar]
  11. B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor, An introduction to the discontinuous Galerkin method for convection-dominated problems, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by Quarteroni. Vol. 1697 of Lect. Notes Math. Springer, Berlin (1998) 151–268. [Google Scholar]
  12. 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 (1989) 90–113. [CrossRef] [MathSciNet] [Google Scholar]
  13. G.H. Golub and C.F. Van Loan, Matrix Computations. Posts and Telecom Press (2011). [Google Scholar]
  14. A. Harten, On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49 (1983a) 151–164. [CrossRef] [Google Scholar]
  15. A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983b) 357–393. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  16. S.-M. Hou and X.-D. Liu, Solutions of multidimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method. J. Sci. Comput. 31 (2007) 127–151. [CrossRef] [Google Scholar]
  17. G.-S. Jiang and C.-W. Shu, On cell entropy inequality for discontinuous Galerkin methods. Math. Comp. 62 (1994) 531–538. [CrossRef] [MathSciNet] [Google Scholar]
  18. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1–26. [Google Scholar]
  19. R. Kress, Numerical analysis. Springer-Verlag (1998). [Google Scholar]
  20. 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, edited by C. de Boor. Academic Press, New York (1974) 89–145. [Google Scholar]
  21. J. Luo, A priori error estimates to Runge–Kutta discontinuous Galerkin finite element method for symmetrizable system of conservation laws with sufficiently smooth solutions. Ph.D. thesis, Nanjing University (2013). [Google Scholar]
  22. S. Osher, Riemann solvers, the entropy condition, and difference approximations. SIAM. J. Numer. Anal. 21 (1984) 217–235. [CrossRef] [MathSciNet] [Google Scholar]
  23. W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory report LA-UR-73-479, Los Alamos, NM (1973). [Google Scholar]
  24. P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357–372. [Google Scholar]
  25. C.W. Schulz-Rinne, Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24 (1993) 76–88. [CrossRef] [MathSciNet] [Google Scholar]
  26. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77 (1988) 439–471. [Google Scholar]
  27. E.F. Toro, Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer (2009). [Google Scholar]
  28. Q. Zhang, Third order explicit Runge–Kutta discontinuous Galerkin method for linear conservation law with inflow boundary condition. J. Sci. Comput. 46 (2010) 294–313. [Google Scholar]
  29. Q. Zhang and C.-W. Shu, Error estimates to smooth solution of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM. J. Numer. Anal. 42 (2004) 641–666. [Google Scholar]
  30. Q. Zhang and C.-W. Shu, Error estimates to smooth solution of Runge–Kutta discontinuous Galerkin methods for symmetrizable system of conservation laws. SIAM. J. Numer. Anal. 44 (2006) 1702–1720. [Google Scholar]
  31. 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. J. Numer. Anal. 48 (2010) 1038–1063. [Google Scholar]
  32. Q. Zhang and C.-W. Shu, Error estimates for the third order explicit Runge–Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data. Numer. Math. 126 (2014) 703–740. [CrossRef] [MathSciNet] [Google Scholar]

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