| Issue |
ESAIM: M2AN
Volume 54, Number 2, March-April 2020
|
|
|---|---|---|
| Page(s) | 705 - 726 | |
| DOI | https://doi.org/10.1051/m2an/2019080 | |
| Published online | 13 March 2020 | |
Research Article
Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements
1
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: chi-wang_shu@brown.edu
Received:
25
July
2019
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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