Error estimates to smooth solutions of Runge-Kutta discontinuous galerkin methods for scalar conservation laws

SIAM Journal on Numerical Analysis

Volumn 42, Issue 2, 2004, Pages 641-666

  Zhang, Qiang ^a,b   Shu, Chi Wang ^c  

^a UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

^b NANKAI UNIVERSITY   (China)

^c BROWN UNIVERSITY   (United States)

Author keywords

Discontinuous Galerkin; Error estimates; Finite element; Total variation diminishing Runge Kutta method

Indexed keywords

DISCONTINUOUS GALERKIN; ERROR ESTIMATES; FINITE ELEMENT; TOTAL VARIATION DIMINISHING RUNGGA-KUTTA METHOD;

APPROXIMATION THEORY; BOUNDARY ELEMENT METHOD; DIFFERENTIAL EQUATIONS; ELECTROMAGNETIC WAVE SCATTERING; FINITE ELEMENT METHOD; INTEGRAL EQUATIONS; MATHEMATICAL OPERATORS; PARAMETER ESTIMATION; RUNGE KUTTA METHODS;

GALERKIN METHODS;

EID: 18444370606     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902404182     Document Type: Article

References (24)

1. S. ADJERID, K. D. DEVINE, J. F. FLAHERTY, AND L. KRIVODONOVA, A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 1097-1112.
2. P. G. CIARLET, Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
3. B. COCKBURN, An introduction to the discontinuous Galerkin method for convection-dominated problems, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Math. 1697, Springer, Berlin, 1998, pp. 151-268.
4. B. COCKBURN AND P.-A. GREMAUD, Error estimates for finite element methods for scalar conservation laws, SIAM J. Numer. Anal., 33 (1996), pp. 522-554.
5. B. COCKBURN, P.-A. GREMAUD, AND J. X. YANG, A priori error estimates for numerical methods for scalar conservation laws Part III: Multidimensional flux-splitting monotone schemes on non-Cartesian grids, SIAM J. Numer. Anal., 35 (1998), pp. 1775-1803.
6. 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), pp. 545-581.
7. 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), pp. 90-113.
8. B. COCKBURN AND C.-W. SHU, TVB Runge-Kutta local projection discontinuous Gakrkin finite element method for scalar conservation laws II: General framework, Math. Comp., 52 (1989), pp. 411-435.
9. B. COCKBURN AND C.-W. SHU, The Runge-Kutta local projection ℙ-discontinuous Galerkin method for scalar conservation laws, RAIRO Model. Math. Anal. Numér., 25 (1991), pp. 337-361.
10. B. COCKBURN AND C.-W. SHU, The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199-224.
11. B. COCKBURN AND C.-W. SHU, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM. J. Numer. Anal., 35 (1998), pp. 2440-2463.
12. B. COCKBURN AND C.-W. SHU, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), pp. 173-261.
13. A. HARTEN, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), pp. 357-393.
14. G.-S. JIANG AND C.-W. SHU, On cell entropy inequality for discontinuous Galerkin methods, Math. Comp., 62 (1994), pp. 531-538.
15. C. JOHNSON AND J. PITKÄRANTA, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), pp. 1-26.
16. H.-O. KREISS, T. A. MANTEUFFEL, B. SWARTZ, B. WENDROFF, AND A. B. WHITE, JR., Supraconvergent schemes on irregular grids, Math. Comp., 47 (1986), pp. 537-554.
17. 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, New York, 1974, pp. 89-145.
18. S. OSHER, Riemann solvers, the entropy condition, and difference approximations, SIAM. J. Numer. Anal., 21 (1984), pp. 217-235.
19. T. E. PETERSON, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal., 28 (1991), pp. 133-140.
20. G. R. RICHTER, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp., 50 (1988), pp. 75-88.
21. C.-W. SHU AND S. OSHER, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77 (1988), pp. 439-471.
22. G. STRANG, Accurate partial difference methods II. Non-linear problems, Numer. Math., 6 (1964), pp. 37-46.
23. L.-A. YING, A second order explicit finite element scheme to multi-dimensional conservation laws and its convergence, Sci. China Ser. A, 43 (2000), pp. 945-957.
24. M. ZHANG AND C.-W. SHU, An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations, Math. Models Methods Appl. Sci., 13 (2003), pp. 395-413.