An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives
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- by Qi Tao, Yan Xu and Chi-Wang Shu HTML | PDF
- Math. Comp. 89 (2020), 2753-2783 Request permission
Abstract:
In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of a local discontinuous Galerkin (LDG) method and the ultraweak discontinuous Galerkin (UWDG) method. First, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply the UWDG method to the system. We first consider the fourth order and fifth order nonlinear PDEs in one space dimension, and then extend our method to general high order problems and two space dimensions. The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables, thereby reducing memory and computational costs. The main advantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs. We prove the stability of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximating its derivatives. A key ingredient in the proof of the error estimates is the construction of the relationship between the derivative and the element interface jump of the numerical solution and the auxiliary variable solution of the solution derivative. With this relationship, we can then use the discrete Sobolev and Poincaré inequalities to obtain the optimal error estimates. The theoretical findings are confirmed by numerical experiments.References
- Susanne C. Brenner, Discrete Sobolev and Poincaré inequalities for piecewise polynomial functions, Electron. Trans. Numer. Anal. 18 (2004), 42–48. MR 2083293
- Susanne C. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal. 41 (2003), no. 1, 306–324. MR 1974504, DOI 10.1137/S0036142902401311
- Yingda Cheng and Chi-Wang Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp. 77 (2008), no. 262, 699–730. MR 2373176, DOI 10.1090/S0025-5718-07-02045-5
- Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
- 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, Guido Kanschat, Ilaria Perugia, and Dominik Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal. 39 (2001), no. 1, 264–285. MR 1860725, DOI 10.1137/S0036142900371544
- 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 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
- Bernardo Cockburn and Chi-Wang Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. MR 1655854, DOI 10.1137/S0036142997316712
- Bo Dong and Chi-Wang Shu, Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems, SIAM J. Numer. Anal. 47 (2009), no. 5, 3240–3268. MR 2551193, DOI 10.1137/080737472
- Jim Douglas Jr. and Todd Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 207–216. MR 0440955
- 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
- Liangyue Ji and Yan Xu, Optimal error estimates of the local discontinuous Galerkin method for Willmore flow of graphs on Cartesian meshes, Int. J. Numer. Anal. Model. 8 (2011), no. 2, 252–283. MR 2740491
- S. M. Han, H Benaroya, and T. Wei, Dynamics of transversely vibrating beams using four engineering theories, Journal of Sound and Vibration, v225 (1999), pp. 935-988.
- J. K. Hunter and J.-M. Vanden-Broeck, Solitary and periodic gravity—capillary waves of finite amplitude, J. Fluid Mech. 134 (1983), 205–219. MR 724024, DOI 10.1017/S0022112083003316
- Hailiang Liu and Peimeng Yin, A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems, J. Sci. Comput. 77 (2018), no. 1, 467–501. MR 3850361, DOI 10.1007/s10915-018-0756-0
- Hailiang Liu and Jue Yan, A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect, J. Comput. Phys. 215 (2006), no. 1, 197–218. MR 2215655, DOI 10.1016/j.jcp.2005.10.016
- 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
- Igor Mozolevski, Endre Süli, and Paulo R. Bösing, $hp$-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation, J. Sci. Comput. 30 (2007), no. 3, 465–491. MR 2295480, DOI 10.1007/s10915-006-9100-1
- W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
- Chi-Wang Shu, Discontinuous Galerkin method for time-dependent problems: survey and recent developments, Recent developments in discontinuous Galerkin finite element methods for partial differential equations, IMA Vol. Math. Appl., vol. 157, Springer, Cham, 2014, pp. 25–62. MR 3203111, DOI 10.1007/978-3-319-01818-8_{2}
- Haijin Wang, Chi-Wang Shu, and Qiang Zhang, Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems, SIAM J. Numer. Anal. 53 (2015), no. 1, 206–227. MR 3296621, DOI 10.1137/140956750
- Haijin Wang, Shiping Wang, Qiang Zhang, and Chi-Wang Shu, Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 4, 1083–1105. MR 3521713, DOI 10.1051/m2an/2015068
- Yinhua Xia, Yan Xu, and Chi-Wang Shu, Efficient time discretization for local discontinuous Galerkin methods, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 677–693. MR 2328730, DOI 10.3934/dcdsb.2007.8.677
- Yan Xu and Chi-Wang Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys. 7 (2010), no. 1, 1–46. MR 2673127, DOI 10.4208/cicp.2009.09.023
- Yan Xu and Chi-Wang Shu, Optimal error estimates of the semidiscrete local discontinuous Galerkin methods for high order wave equations, SIAM J. Numer. Anal. 50 (2012), no. 1, 79–104. MR 2888305, DOI 10.1137/11082258X
- Jue Yan and Chi-Wang Shu, A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal. 40 (2002), no. 2, 769–791. MR 1921677, DOI 10.1137/S0036142901390378
- Jue Yan and Chi-Wang Shu, Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), 2002, pp. 27–47. MR 1910550, DOI 10.1023/A:1015132126817
Additional Information
- Qi Tao
- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
- Email: taoq@mail.ustc.edu.cn
- Yan Xu
- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
- Email: yxu@ustc.edu.cn
- Chi-Wang Shu
- Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 242268
- Email: chi-wang_shu@brown.edu
- Received by editor(s): September 2, 2019
- Received by editor(s) in revised form: February 21, 2020
- Published electronically: August 4, 2020
- Additional Notes: The research of the first author was supported by the China Scholarship Council.
The research of the second author was supported by National Numerical Windtunnel grants NNW2019ZT4-B08, Science Challenge Project TZZT2019-A2.3, and NSFC grants 11722112.
The second author is the corresponding author.
The research of the third author was supported by NSF grant DMS-1719410. - © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 89 (2020), 2753-2783
- MSC (2010): Primary 65M60; Secondary 35G25
- DOI: https://doi.org/10.1090/mcom/3562
- MathSciNet review: 4136546