A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws

Luo, Juan; Shu, Chi-Wang; Zhang, Qiang

doi:10.1051/m2an/2014063

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

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 991-1018.

Résumé

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 $k \geq 2$ . Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition $τ \leq γ h$ , 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 : 2013-08-22

MR Zbl | 1 citation dans Numdam

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

Affiliations des auteurs :

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

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     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},
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     zbl = {1327.65193},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2014063/}
}

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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/

Bibliographie
Cité par

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 P $^{1}$ -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

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