Volume 55, Number 1, January-February 2021
|Page(s)||329 - 356|
|Published online||18 February 2021|
Central discontinuous Galerkin methods on overlapping meshes for wave equations
Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China
2 South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Canton 510631, Guangdong, P.R. China
3 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
4 School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026 Anhui, P.R. China
* Corresponding author: firstname.lastname@example.org
Accepted: 17 September 2020
In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise Pk elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k ≥ 0. In particular, we adopt the techniques in Liu et al. (SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.
Mathematics Subject Classification: 65M60 / 65M12 / 65M15
Key words: Optimal error estimates / central DG method / second order wave equation / dispersion analysis
© EDP Sciences, SMAI 2021
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