no. 2
Research Article
Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using P k elements
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 2, pp. 705-726

In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using P$$ elements on uniform Cartesian meshes, and prove that the error in the L2 norm achieves optimal (k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.

DOI : 10.1051/m2an/2019080
Classification : 65M60, 65M15
Keywords: Optimal error estimate, discontinuous Galerkin method, upwind fluxes 
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     author = {Liu, Yong and Shu, Chi-Wang and Zhang, Mengping},
     title = {Optimal error estimates of the semidiscrete discontinuous {Galerkin} methods for two dimensional hyperbolic equations on {Cartesian} meshes using $P^k$ elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {705--726},
     year = {2020},
     publisher = {EDP Sciences},
     volume = {54},
     number = {2},
     doi = {10.1051/m2an/2019080},
     mrnumber = {4076058},
     zbl = {1439.65115},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2019080/}
}
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Liu, Yong; Shu, Chi-Wang; Zhang, Mengping. Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using $P^k$ elements. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 2, pp. 705-726. doi: 10.1051/m2an/2019080

W. Cao, C.-W. Shu, Y. Yang and Z. Zhang, Superconvergence of discontinuous Galerkin method for scalar nonlinear hyperbolic equations. SIAM J. Numer. Anal. 56 (2018) 732–765. | MR | Zbl | DOI

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

B. Cockburn and C.-W. Shu, TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52 (1989) 411–435. | MR | Zbl

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2465. | MR | Zbl | DOI

B. Cockburn and C.-W. Shu, The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141 (1998) 199–224. | MR | Zbl | DOI

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. | MR | Zbl | DOI

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. Math. Comput. 54 (1990) 545–581. | MR | Zbl

B. Cockburn, B. Dong and J. Guzmán, Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46 (2008) 1250–1265. | MR | Zbl | DOI

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. | MR | Zbl | DOI

Y. Liu, C.-W. Shu and M. Zhang, Optimal error estimates of the semidiscrete central discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 56 (2018) 520–541. | MR | Zbl | DOI

X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput. 85 (2016) 1225–1261. | MR | Zbl | DOI

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

W.H. Reed and T.R. Hill, Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Scientific Laboratory report LA-UR-74-479, Los Alamos, NM (1973).

G.R. Richter, An optimal-order error estimate for the discontinuous Galerkin method. Math. Comput. 50 (1988) 75–88. | MR | Zbl | DOI

Y. Xu and C.-W. Shu, Optimal error estimates of the semidiscrete local discontinuous galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50 (2012) 79–104. | MR | Zbl | DOI

Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42 (2004) 641–666. | MR | Zbl | DOI

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