Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes

Sun, Zheng; Xing, Yulong

doi:10.1090/mcom/3605

Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes
HTML articles powered by AMS MathViewer

by Zheng Sun and Yulong Xing HTML | PDF

Math. Comp. 90 (2021), 1741-1772 Request permission

Abstract:

$L^2$ stable discontinuous Galerkin method with a family of numerical fluxes was studied for the one-dimensional wave equation by Cheng, Chou, Li, and Xing in [Math. Comp. 86 (2017), pp. 121–155]. Although optimal convergence rates were numerically observed with wide choices of parameters in the numerical fluxes, their error estimates were only proved for a sub-family with the construction of a local projection. In this paper, we first complete the one-dimensional analysis by providing optimal error estimates that match all numerical observations in that paper. The key ingredient is to construct an optimal global projection with the characteristic decomposition. We then extend the analysis on optimal error estimate to multidimensions by constructing a global projection on unstructured meshes, which can be considered as a perturbation away from the local projection studied by Cockburn, Gopalakrishnan, and Sayas in [Math. Comp. 79 (2010), pp. 1351–1367] for hybridizable discontinuous Galerkin methods. As a main contribution, we use a novel energy argument to prove the optimal approximation property of the global projection. This technique does not require explicit assembly of the matrix for the perturbed terms and hence can be easily used for unstructured meshes in multidimensions. Finally, numerical tests in two dimensions are provided to validate our analysis is sharp and at least one of the unknowns will degenerate to suboptimal rates if the assumptions are not satisfied.

References

Mark Ainsworth and Guosheng Fu, Dispersive behavior of an energy-conserving discontinuous Galerkin method for the one-way wave equation, J. Sci. Comput. 79 (2019), no. 1, 209–226. MR 3936245, DOI 10.1007/s10915-018-0846-z
M. Ainsworth, P. Monk, and W. Muniz, Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation, J. Sci. Comput. 27 (2006), no. 1-3, 5–40. MR 2285764, DOI 10.1007/s10915-005-9044-x
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys. 131 (1997), no. 2, 267–279. MR 1433934, DOI 10.1006/jcph.1996.5572
J. L. Bona, H. Chen, O. Karakashian, and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Math. Comp. 82 (2013), no. 283, 1401–1432. MR 3042569, DOI 10.1090/S0025-5718-2013-02661-0
Yingda Cheng, Ching-Shan Chou, Fengyan Li, and Yulong Xing, $L^2$ stable discontinuous Galerkin methods for one-dimensional two-way wave equations, Math. Comp. 86 (2017), no. 303, 121–155. MR 3557796, DOI 10.1090/mcom/3090
Yao Cheng, Xiong Meng, and Qiang Zhang, Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp. 86 (2017), no. 305, 1233–1267. MR 3614017, DOI 10.1090/mcom/3141
Ching-Shan Chou, Chi-Wang Shu, and Yulong Xing, Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media, J. Comput. Phys. 272 (2014), 88–107. MR 3212263, DOI 10.1016/j.jcp.2014.04.009
Eric T. Chung and Björn Engquist, Optimal discontinuous Galerkin methods for wave propagation, SIAM J. Numer. Anal. 44 (2006), no. 5, 2131–2158. MR 2263043, DOI 10.1137/050641193
Bernardo Cockburn and Bo Dong, An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems, J. Sci. Comput. 32 (2007), no. 2, 233–262. MR 2320571, DOI 10.1007/s10915-007-9130-3
Bernardo Cockburn, Zhixing Fu, Allan Hungria, Liangyue Ji, Manuel A. Sánchez, and Francisco-Javier Sayas, Stormer-Numerov HDG methods for acoustic waves, J. Sci. Comput. 75 (2018), no. 2, 597–624. MR 3780780, DOI 10.1007/s10915-017-0547-z
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351–1367. MR 2629996, DOI 10.1090/S0025-5718-10-02334-3
Bernardo Cockburn, Suchung Hou, and Chi-Wang Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp. 54 (1990), no. 190, 545–581. MR 1010597, DOI 10.1090/S0025-5718-1990-1010597-0
Bernardo Cockburn, San Yih Lin, and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113. MR 1015355, DOI 10.1016/0021-9991(89)90183-6
Bernardo Cockburn and Vincent Quenneville-Bélair, Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation, Math. Comp. 83 (2014), no. 285, 65–85. MR 3120582, DOI 10.1090/S0025-5718-2013-02743-3
Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI 10.1090/S0025-5718-1989-0983311-4
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 3, 337–361 (English, with French summary). MR 1103092, DOI 10.1051/m2an/1991250303371
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. MR 1619652, DOI 10.1006/jcph.1998.5892
Gary C. Cohen, Higher-order numerical methods for transient wave equations, Scientific Computation, Springer-Verlag, Berlin, 2002. With a foreword by R. Glowinski. MR 1870851, DOI 10.1007/978-3-662-04823-8
Richard S. Falk and Gerard R. Richter, Explicit finite element methods for symmetric hyperbolic equations, SIAM J. Numer. Anal. 36 (1999), no. 3, 935–952. MR 1688992, DOI 10.1137/S0036142997329463
Guosheng Fu and Chi-Wang Shu, Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems, J. Comput. Phys. 394 (2019), 329–363. MR 3960714, DOI 10.1016/j.jcp.2019.05.050
Pei Fu, Yingda Cheng, Fengyan Li, and Yan Xu, Discontinuous Galerkin methods with optimal $L^2$ accuracy for one dimensional linear PDEs with high order spatial derivatives, J. Sci. Comput. 78 (2019), no. 2, 816–863. MR 3918671, DOI 10.1007/s10915-018-0788-5
Marcus J. Grote, Anna Schneebeli, and Dominik Schötzau, Discontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal. 44 (2006), no. 6, 2408–2431. MR 2272600, DOI 10.1137/05063194X
Guang Shan Jiang and Chi-Wang Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comp. 62 (1994), no. 206, 531–538. MR 1223232, DOI 10.1090/S0025-5718-1994-1223232-7
Jia Li, Dazhi Zhang, Xiong Meng, Boying Wu, and Qiang Zhang, Discontinuous Galerkin methods for nonlinear scalar conservation laws: generalized local Lax-Friedrichs numerical fluxes, SIAM J. Numer. Anal. 58 (2020), no. 1, 1–20. MR 4046779, DOI 10.1137/19M1243798
Hailiang Liu and Nattapol Ploymaklam, A local discontinuous Galerkin method for the Burgers-Poisson equation, Numer. Math. 129 (2015), no. 2, 321–351. MR 3300422, DOI 10.1007/s00211-014-0641-1
Minghui Liu, Boying Wu, and Xiong Meng, Optimal error estimates of the discontinuous Galerkin method with upwind-biased fluxes for 2D linear variable coefficients hyperbolic equations, J. Sci. Comput. 83 (2020), no. 1, Paper No. 9, 19. MR 4079266, DOI 10.1007/s10915-020-01197-x
Yong Liu, Jianfang Lu, Chi-Wang Shu, and Mengping Zhang, Central discontinuous Galerkin methods on overlapping meshes for wave equations, ESAIM Math. Model. Numer. Anal. 55 (2021), no. 1, 329–356. MR 4216835, DOI 10.1051/m2an/2020069
Y. Liu, C.-W. Shu, and M. Zhang, Sub-optimal convergence of discontinuous Galerkin methods with central fluxes for even degree polynomial approximations, Journal of Computational Mathematics, to appear.
Xiong Meng, Chi-Wang Shu, and Boying Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations, Math. Comp. 85 (2016), no. 299, 1225–1261. MR 3454363, DOI 10.1090/mcom/3022
Peter Monk and Gerard R. Richter, A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, J. Sci. Comput. 22/23 (2005), 443–477. MR 2142205, DOI 10.1007/s10915-004-4132-5
N. C. Nguyen, J. Peraire, and B. Cockburn, High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics, J. Comput. Phys. 230 (2011), no. 10, 3695–3718. MR 2783813, DOI 10.1016/j.jcp.2011.01.035
W. H. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Technical report, Los Alamos Scientific Lab., N. Mex.(USA), 1973.
M. A. Sánchez, C. Ciuca, N. C. Nguyen, J. Peraire, and B. Cockburn, Symplectic Hamiltonian HDG methods for wave propagation phenomena, J. Comput. Phys. 350 (2017), 951–973. MR 3707190, DOI 10.1016/j.jcp.2017.09.010
J. Schöberl, NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Computing and visualization in science, 1(1):41–52, 1997.
M. Stanglmeier, N. C. Nguyen, J. Peraire, and B. Cockburn, An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation, Comput. Methods Appl. Mech. Engrg. 300 (2016), 748–769. MR 3452794, DOI 10.1016/j.cma.2015.12.003
Zheng Sun and Chi-Wang Shu, Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 3, 1063–1087. MR 3666657, DOI 10.1051/m2an/2016049
Zheng Sun and Chi-Wang Shu, Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations, Ann. Math. Sci. Appl. 2 (2017), no. 2, 255–284. MR 3690156, DOI 10.4310/AMSA.2017.v2.n2.a3
Zheng Sun and Chi-wang Shu, Strong stability of explicit Runge-Kutta time discretizations, SIAM J. Numer. Anal. 57 (2019), no. 3, 1158–1182. MR 3953465, DOI 10.1137/18M122892X
Z. Sun and C.-W. Shu. Error analysis of Runge–Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations. arXiv preprint arXiv:2001.00971, 2020.
Zheng Sun and Yulong Xing, On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: energy conservation and multi-symplecticity, J. Comput. Phys. 419 (2020), 109662, 25. MR 4116802, DOI 10.1016/j.jcp.2020.109662
Yulong Xing, Ching-Shan Chou, and Chi-Wang Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Probl. Imaging 7 (2013), no. 3, 967–986. MR 3105364, DOI 10.3934/ipi.2013.7.967
Yuan Xu, Chi-Wang Shu, and Qiang Zhang, Error estimate of the fourth-order Runge-Kutta discontinuous Galerkin methods for linear hyperbolic equations, SIAM J. Numer. Anal. 58 (2020), no. 5, 2885–2914. MR 4161755, DOI 10.1137/19M1280077
Qiang Zhang and Chi-Wang Shu, Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal. 48 (2010), no. 3, 1038–1063. MR 2669400, DOI 10.1137/090771363