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
Mots-clés : 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},
     publisher = {EDP-Sciences},
     volume = {54},
     number = {2},
     year = {2020},
     doi = {10.1051/m2an/2019080},
     mrnumber = {4076058},
     zbl = {1439.65115},
     language = {en},
     url = {http://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. http://www.numdam.org/articles/10.1051/m2an/2019080/

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