Volume 54, Number 2, March-April 2020
|Page(s)||705 - 726|
|Published online||13 March 2020|
Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements
School of Mathematical Sciences, University of Science and Technology of China, 230026 Hefei, Anhui, P.R. China
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
* Corresponding author: email@example.com
Accepted: 19 November 2019
In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using Pk 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.
Mathematics Subject Classification: 65M60 / 65M15
Key words: Optimal error estimate / discontinuous Galerkin method / upwind fluxes
© EDP Sciences, SMAI 2020
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