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$$ 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.
Keywords: Optimal error estimates, central DG method, second order wave equation, dispersion analysis
@article{M2AN_2021__55_1_329_0,
author = {Liu, Yong and Lu, Jianfang and Shu, Chi-Wang and Zhang, Mengping},
title = {Central discontinuous {Galerkin} methods on overlapping meshes for wave equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {329--356},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {1},
doi = {10.1051/m2an/2020069},
mrnumber = {4216835},
zbl = {1470.65168},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020069/}
}
TY - JOUR AU - Liu, Yong AU - Lu, Jianfang AU - Shu, Chi-Wang AU - Zhang, Mengping TI - Central discontinuous Galerkin methods on overlapping meshes for wave equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 329 EP - 356 VL - 55 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020069/ DO - 10.1051/m2an/2020069 LA - en ID - M2AN_2021__55_1_329_0 ER -
%0 Journal Article %A Liu, Yong %A Lu, Jianfang %A Shu, Chi-Wang %A Zhang, Mengping %T Central discontinuous Galerkin methods on overlapping meshes for wave equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 329-356 %V 55 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020069/ %R 10.1051/m2an/2020069 %G en %F M2AN_2021__55_1_329_0
Liu, Yong; Lu, Jianfang; Shu, Chi-Wang; Zhang, Mengping. Central discontinuous Galerkin methods on overlapping meshes for wave equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 1, pp. 329-356. doi: 10.1051/m2an/2020069
[1] , A local discontinuous Galerkin method for the second-order wave equation. Comput. Methods Appl. Mech. Eng. 209 (2012) 129–143. | MR | Zbl | DOI
[2] , Numerical Methods for Ordinary Differential Equations, 2nd edition. John Wiley & Sons (2008). | MR | Zbl | DOI
[3] , , and , stable discontinuous Galerkin methods for one dimensional two-way wave equations. Math. Comput. 86 (2017) 121–155. | MR | Zbl | DOI
[4] and , Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47 (2009) 3820–3848. | MR | Zbl | DOI
[5] , The Finite Element Method for Elliptic Problems. North Holland (1978). | MR | Zbl
[6] and , TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. general framework. Math. Comput. 52 (1989) 411–435. | MR | Zbl
[7] and , The Runge-Kutta local projection -discontinuous Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. 25 (1991) 337–361. | MR | Zbl | Numdam | DOI
[8] and , The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141 (1998) 199–224. | MR | Zbl | DOI
[9] , and , TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84 (1989) 90–113. | MR | Zbl | DOI
[10] , and , The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54 (1990) 545–581. | MR | Zbl
[11] , , and , Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39 (2001) 264–285. | MR | Zbl | DOI
[12] , Higher-order numerical methods for transient wave equations. In: Scientific Computation, Springer-Verlag, Berlin (2002). With a foreword by R. Glowinski. | MR | Zbl | DOI
[13] , Numerical methods for wave equations in geophysical fluid dynamics. In: Vol. 32 of Texts in Applied Mathematics. Springer, New York (1999). | MR | Zbl | DOI
[14] and , Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128 (2001) 83–131. | MR | Zbl | DOI
[15] and , Numerical analysis of spectral methods: theory and applications. In: CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. Society for Industrial and Applied Mathematics, Philadelphia, PA (1977). | MR | Zbl
[16] , and , Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. | MR | Zbl | DOI
[17] , and , Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408–2431. | MR | Zbl | DOI
[18] , and , Effective Computational Methods for Wave Propagation. Chapman & Hall/CRC (2008). | MR | Zbl | DOI
[19] , Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2002). | MR | Zbl | DOI
[20] , , and , Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45 (2007) 2442–2467. | MR | Zbl | DOI
[21] , , and , stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM: M2AN 42 (2008) 593–607. | MR | Zbl | Numdam | DOI
[22] , and , Optimal error estimates of the semidiscrete central discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 56 (2018) 520–541. | MR | Zbl | DOI
[23] , and , Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using elements. ESAIM: M2AN 54 (2020) 705–726. | MR | Zbl | Numdam | DOI
[24] , and , Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput. 85 (2016) 1225–1261. | MR | Zbl | DOI
[25] and , Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory report LA-UR-73-479 (1973).
[26] and , Operator bounds and time step conditions for DG and central DG methods. J. Sci. Comput. 62 (2015) 532–554. | MR | Zbl | DOI
[27] , and , Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Prob. Imaging 7 (2013) 967–986. | MR | Zbl | DOI
[28] and , Optimal error estimates of the semidiscrete local discontinuous galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50 (2012) 79–104. | MR | Zbl | DOI
[29] and , New central and central discontinuous Galerkin schemes on overlapping cells of unstructured grids for solving ideal magnetohydrodynamic equations with globally divergence-free magnetic field. J. Comput. Phys. 327 (2016) 203–224. | MR | Zbl | DOI
[30] , and , Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension. Math. Comput. 89 (2020) 2113–2139. | MR | Zbl | DOI
Cité par Sources :
