Central discontinuous Galerkin methods on overlapping meshes for wave equations
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 P k 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 P 1 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.
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10225997
Journal Name:
ESAIM: Mathematical Modelling and Numerical Analysis
Volume:
55
Issue:
1
Page Range or eLocation-ID:
329 to 356
ISSN:
0764-583X
1. In this paper, we construct, analyze, and numerically validate conservative discontinuous Galerkin (DG) schemes for approximating the Schr\"{o}dinger-Poisson equation. The proposed schemes all satisfy both mass and energy conservation. For the semi-discrete DG scheme optimal $L^2$ error estimates are obtained. Efficient iterative solvers are also constructed to solve the second order implicit time discretization. A number of numerical tests are presented to demonstrate the method’s accuracy and robustness, confirming that both mass and energy are well preserved over long time simulations.