A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 991-1018.

In this paper we present an a priori error estimate of the Runge–Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge–Kutta method and the finite element space is made up of piecewise polynomials of degree k2. Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition τγh, where h is the element length and τ is time step, and γ is a positive constant independent of h and τ. Optimal estimates are also considered when the upwind numerical flux is used.

Reçu le :
DOI : 10.1051/m2an/2014063
Classification : 65M60, 65M12
Mots-clés : Discontinuous Galerkin method, Runge–Kutta method, error estimates, symmetrizable system of conservation laws, energy analysis
Luo, Juan 1 ; Shu, Chi-Wang 2 ; Zhang, Qiang 1

1 Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu Province, P.R. China
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
@article{M2AN_2015__49_4_991_0,
     author = {Luo, Juan and Shu, Chi-Wang and Zhang, Qiang},
     title = {A priori error estimates to smooth solutions of the third order {Runge{\textendash}Kutta} discontinuous {Galerkin} method for symmetrizable systems of conservation laws},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {991--1018},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {4},
     year = {2015},
     doi = {10.1051/m2an/2014063},
     mrnumber = {3371901},
     zbl = {1327.65193},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2014063/}
}
TY  - JOUR
AU  - Luo, Juan
AU  - Shu, Chi-Wang
AU  - Zhang, Qiang
TI  - A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 991
EP  - 1018
VL  - 49
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2014063/
DO  - 10.1051/m2an/2014063
LA  - en
ID  - M2AN_2015__49_4_991_0
ER  - 
%0 Journal Article
%A Luo, Juan
%A Shu, Chi-Wang
%A Zhang, Qiang
%T A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 991-1018
%V 49
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2014063/
%R 10.1051/m2an/2014063
%G en
%F M2AN_2015__49_4_991_0
Luo, Juan; Shu, Chi-Wang; Zhang, Qiang. A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 991-1018. doi : 10.1051/m2an/2014063. http://www.numdam.org/articles/10.1051/m2an/2014063/

R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

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. | DOI | MR | Zbl

P.G. Ciarlet, Finite Element Method for Elliptic Problems. North–Holland, Amsterdam (1978). | MR | Zbl

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. | DOI | MR | Zbl

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. | MR | Zbl

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. | DOI | Numdam | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

B. Cockburn and C.-W. Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems J. Sci. Comput. 16 (2001) 173–261. | DOI | MR | Zbl

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. | MR | Zbl

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. | MR | Zbl

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. | DOI | MR | Zbl

G.H. Golub and C.F. Van Loan, Matrix Computations. Posts and Telecom Press (2011).

A. Harten, On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49 (1983a) 151–164. | DOI | MR | Zbl

A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983b) 357–393. | DOI | MR | Zbl

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. | DOI | MR | Zbl

G.-S. Jiang and C.-W. Shu, On cell entropy inequality for discontinuous Galerkin methods. Math. Comp. 62 (1994) 531–538. | DOI | MR | Zbl

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46 (1986) 1–26. | DOI | MR | Zbl

R. Kress, Numerical analysis. Springer-Verlag (1998). | MR | Zbl

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. | MR | Zbl

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).

S. Osher, Riemann solvers, the entropy condition, and difference approximations. SIAM. J. Numer. Anal. 21 (1984) 217–235. | DOI | MR | Zbl

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).

P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357–372. | DOI | MR | Zbl

C.W. Schulz-Rinne, Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24 (1993) 76–88. | DOI | MR | Zbl

C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77 (1988) 439–471. | DOI | MR | Zbl

E.F. Toro, Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer (2009). | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

  • Zhang, Hongjuan; Wu, Boying; Meng, Xiong Analysis of the local discontinuous Galerkin method with generalized fluxes for one-dimensional nonlinear convection-diffusion systems, Science China. Mathematics, Volume 66 (2023) no. 11, pp. 2641-2664 | DOI:10.1007/s11425-022-2035-y | Zbl:1528.65085
  • Sun, Jiawei; Xie, Shusen; Xing, Yulong Local discontinuous Galerkin methods for the abcd nonlinear Boussinesq system, Communications on Applied Mathematics and Computation, Volume 4 (2022) no. 2, pp. 381-416 | DOI:10.1007/s42967-021-00119-4 | Zbl:1499.65528
  • Jiao, Mengjiao; Jiang, Yan; Shu, Chi-Wang; Zhang, Mengping Optimal error estimates to smooth solutions of the central discontinuous Galerkin methods for nonlinear scalar conservation laws, European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis, Volume 56 (2022) no. 4, pp. 1401-1435 | DOI:10.1051/m2an/2022037 | Zbl:1497.65174
  • Valbuena, Sonia; Vega, Carlos A. Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model, Mathematics, Volume 10 (2022) no. 18, p. 3314 | DOI:10.3390/math10183314
  • Ai, Jingqi; Xu, Yuan; Shu, Chi-Wang; Zhang, Qiang L2 error estimate to smooth solutions of high order Runge-Kutta discontinuous Galerkin method for scalar nonlinear conservation laws with and without sonic points, SIAM Journal on Numerical Analysis, Volume 60 (2022) no. 4, pp. 1741-1773 | DOI:10.1137/21m1435495 | Zbl:1501.65062
  • Zhang, Chao; Xu, Yan; Xia, Yinhua Local discontinuous Galerkin methods to a dispersive system of KdV-type equations, Journal of Scientific Computing, Volume 86 (2021) no. 1, p. 43 (Id/No 4) | DOI:10.1007/s10915-020-01370-2 | Zbl:1459.35336
  • Du, Qiang; Ju, Lili; Lu, Jianfang Analysis of fully discrete approximations for dissipative systems and application to time-dependent nonlocal diffusion problems, Journal of Scientific Computing, Volume 78 (2019) no. 3, pp. 1438-1466 | DOI:10.1007/s10915-018-0815-6 | Zbl:1419.65058
  • Fu, Pei; Schnücke, Gero; Xia, Yinhua Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes, Mathematics of Computation, Volume 88 (2019) no. 319, pp. 2221-2255 | DOI:10.1090/mcom/3417 | Zbl:1462.65140
  • Meng, Xiong; Ryan, Jennifer K Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws, IMA Journal of Numerical Analysis, Volume 38 (2018) no. 1, p. 125 | DOI:10.1093/imanum/drw072
  • Puelz, Charles; Rivière, Béatrice A priori error estimates of Adams-Bashforth discontinuous Galerkin methods for scalar nonlinear conservation laws, Journal of Numerical Mathematics, Volume 26 (2018) no. 3, pp. 151-172 | DOI:10.1515/jnma-2017-0011 | Zbl:1407.65161
  • Puelz, Charles; Čanić, Sunčica; Rivière, Béatrice; Rusin, Craig G. Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods, Applied Numerical Mathematics, Volume 115 (2017), pp. 114-141 | DOI:10.1016/j.apnum.2017.01.005 | Zbl:1361.92020
  • Sun, Zheng; Shu, Chi-Wang Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 51 (2017) no. 3, p. 1063 | DOI:10.1051/m2an/2016049
  • Cockburn, Bernardo Discontinuous Galerkin Methods for Computational Fluid Dynamics, Encyclopedia of Computational Mechanics Second Edition (2017), p. 1 | DOI:10.1002/9781119176817.ecm2053
  • Huang, Juntao; Shu, Chi-Wang A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr-Debye model, M3AS. Mathematical Models Methods in Applied Sciences, Volume 27 (2017) no. 3, pp. 549-579 | DOI:10.1142/s0218202517500099 | Zbl:1360.65235
  • Shu, Chi-Wang Discontinuous Galerkin methods for time-dependent convection dominated problems: basics, recent developments and comparison with other methods, Building bridges: connections and challenges in modern approaches to numerical partial differential equations. Selected papers based on the presentations at the 101st LMS-EPSRC symposium, Durham, UK, July 8–16, 2014, Cham: Springer, 2016, pp. 371-399 | DOI:10.1007/978-3-319-41640-3_12 | Zbl:1357.65179
  • Qiu, J.; Zhang, Q. Stability, Error Estimate and Limiters of Discontinuous Galerkin Methods, Handbook of Numerical Methods for Hyperbolic Problems - Basic and Fundamental Issues, Volume 17 (2016), p. 147 | DOI:10.1016/bs.hna.2016.06.001
  • Liu, YunXian; Shu, Chi-Wang Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices, Science China. Mathematics, Volume 59 (2016) no. 1, pp. 115-140 | DOI:10.1007/s11425-015-5055-8 | Zbl:1342.65183

Cité par 17 documents. Sources : Crossref, zbMATH