Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

Yang, Yang; Cai, Xiaofeng; Qiu, Jing-Mei

doi:10.1090/mcom/3527

Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension
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Math. Comp. 89 (2020), 2113-2139 Request permission

Abstract:

In this paper, we apply semi-Lagrangian discontinuous Galerkin (SLDG) methods for linear hyperbolic equations in one space dimension and analyze the error between the numerical and exact solutions under the $L^2$-norm. In all the previous works, the theoretical analysis of the SLDG method would suggest a suboptimal convergence rate due to the error accumulation over time steps. However, numerical experiments demonstrate an optimal convergence rate and, if the terminal time is large, a superconvergence rate. In this paper, we will prove optimal convergence and optimal superconvergence rates. There are three main difficulties: 1. The error analysis on overlapping meshes. Due to the nature of the semi-Lagrangian time discretization, we need to introduce the background Eulerian mesh and the shifted mesh. The two meshes are staggered, and it is not easy to construct local projections and to handle the error accumulation during time evolution. 2. The superconvergence of time-dependent terms under the $L^2$-norm. The error of the numerical and exact solutions can be divided into two parts, the projection error and the time-dependent superconvergence term. The projection strongly depends on the superconvergence rates. Therefore, we need to construct a sequence of projections and gradually improve the superconvergence rates. 3. The stopping criterion of the sequence of projections. The sequence of projections are basically of the same form. We need to show that the projections exist up to some certain order since the superconvergence rate cannot be infinity. Hence, we will seek some “hidden” condition for the existence of the projections. In this paper, we will solve all the three difficulties and construct several local projections to prove the optimal convergence and superconvergence rates. Numerical experiments verify the theoretical findings.

References

Slimane Adjerid and Thomas C. Massey, Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 25-28, 3331–3346. MR 2220922, DOI 10.1016/j.cma.2005.06.017
Slimane Adjerid and Thomas Weinhart, Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 37-40, 3113–3129. MR 2567860, DOI 10.1016/j.cma.2009.05.016
Slimane Adjerid and Thomas Weinhart, Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems, Math. Comp. 80 (2011), no. 275, 1335–1367. MR 2785461, DOI 10.1090/S0025-5718-2011-02460-9
Xiaofeng Cai, Wei Guo, and Jing-Mei Qiu, A high order conservative semi-Lagrangian discontinuous Galerkin method for two-dimensional transport simulations, J. Sci. Comput. 73 (2017), no. 2-3, 514–542. MR 3719597, DOI 10.1007/s10915-017-0554-0
Xiaofeng Cai, Wei Guo, and Jing-Mei Qiu, A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting, J. Comput. Phys. 354 (2018), 529–551. MR 3738121, DOI 10.1016/j.jcp.2017.10.048
Waixiang Cao, Dongfang Li, Yang Yang, and Zhimin Zhang, Superconvergence of discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 2, 467–486. MR 3626407, DOI 10.1051/m2an/2016026
Waixiang Cao, Chi-Wang Shu, and Zhimin Zhang, Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 6, 2213–2235. MR 3745170, DOI 10.1051/m2an/2017026
Waixiang Cao, Chi-Wang Shu, Yang Yang, and Zhimin Zhang, Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations, SIAM J. Numer. Anal. 53 (2015), no. 4, 1651–1671. MR 3365565, DOI 10.1137/140996203
Waixiang Cao, Chi-Wang Shu, Yang Yang, and Zhimin Zhang, Superconvergence of discontinuous Galerkin method for scalar nonlinear hyperbolic equations, SIAM J. Numer. Anal. 56 (2018), no. 2, 732–765. MR 3780748, DOI 10.1137/17M1128605
Waixiang Cao and Zhimin Zhang, Some recent developments in superconvergence of discontinuous Galerkin methods for time-dependent partial differential equations, J. Sci. Comput. 77 (2018), no. 3, 1402–1423. MR 3874779, DOI 10.1007/s10915-018-0762-2
Waixiang Cao, Zhimin Zhang, and Qingsong Zou, Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations, SIAM J. Numer. Anal. 52 (2014), no. 5, 2555–2573. MR 3270187, DOI 10.1137/130946873
N. Chalmers and L. Krivodonova, Spatial and modal superconvergence of the discontinuous Galerkin method for linear equations, J. Sci. Comput. 72 (2017), no. 1, 128–146. MR 3661100, DOI 10.1007/s10915-016-0349-8
Yingda Cheng and Chi-Wang Shu, Superconvergence and time evolution of discontinuous Galerkin finite element solutions, J. Comput. Phys. 227 (2008), no. 22, 9612–9627. MR 2467636, DOI 10.1016/j.jcp.2008.07.010
Yingda Cheng and Chi-Wang Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM J. Numer. Anal. 47 (2010), no. 6, 4044–4072. MR 2585178, DOI 10.1137/090747701
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Lukas Einkemmer and Alexander Ostermann, Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov-Poisson equations, SIAM J. Numer. Anal. 52 (2014), no. 2, 757–778. MR 3187672, DOI 10.1137/120898620
Maurizio Falcone and Roberto Ferretti, Convergence analysis for a class of high-order semi-Lagrangian advection schemes, SIAM J. Numer. Anal. 35 (1998), no. 3, 909–940. MR 1619910, DOI 10.1137/S0036142994273513
Li Guo and Yang Yang, Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations with singular initial data, Int. J. Numer. Anal. Model. 14 (2017), no. 3, 342–354. MR 3652168
Wei Guo and Jing-Mei Qiu, Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation, J. Comput. Phys. 234 (2013), 108–132. MR 2999770, DOI 10.1016/j.jcp.2012.09.014
W. Guo and J.-M. Qiu, A conservative semi-Lagrangian discontinuous Galerkin scheme on the cubed sphere, Mon. Weather Rev. 142 (2014), 457-475.
Wei Guo, Xinghui Zhong, and Jing-Mei Qiu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach, J. Comput. Phys. 235 (2013), 458–485. MR 3017607, DOI 10.1016/j.jcp.2012.10.020
Yong Liu, Chi-Wang Shu, and Mengping Zhang, Optimal error estimates of the semidiscrete central discontinuous Galerkin methods for linear hyperbolic equations, SIAM J. Numer. Anal. 56 (2018), no. 1, 520–541. MR 3763093, DOI 10.1137/16M1089484
Yong Liu, Chi-Wang Shu, and Mengping Zhang, Superconvergence of energy-conserving discontinuous Galerkin methods for linear hyperbolic equations, Commun. Appl. Math. Comput. 1 (2019), no. 1, 101–116. MR 4022325, DOI 10.1007/s42967-019-0006-y
X. Meng, C.-W. Shu, and Y. Yang, Superconvergence of discontinuous Galerkin methods for time-dependent partial differential equations, Sci. Sin. Math., 45 (2015), 1041-1060.
Xiong Meng, Chi-Wang Shu, Qiang Zhang, and Boying Wu, Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension, SIAM J. Numer. Anal. 50 (2012), no. 5, 2336–2356. MR 3022221, DOI 10.1137/110857635
Jing-Mei Qiu and Chi-Wang Shu, Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov-Poisson system, J. Comput. Phys. 230 (2011), no. 23, 8386–8409. MR 2843721, DOI 10.1016/j.jcp.2011.07.018
James A. Rossmanith and David C. Seal, A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations, J. Comput. Phys. 230 (2011), no. 16, 6203–6232. MR 2806222, DOI 10.1016/j.jcp.2011.04.018
Philip Roe, A simple explanation of superconvergence for discontinuous Galerkin solutions to $u_t+u_x=0$, Commun. Comput. Phys. 21 (2017), no. 4, 905–912. MR 3621618, DOI 10.4208/cicp.OA-2016-0052
Yang Yang and Chi-Wang Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM J. Numer. Anal. 50 (2012), no. 6, 3110–3133. MR 3022256, DOI 10.1137/110857647
Jianxian Qiu and Chi-Wang Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method. II. Two dimensional case, Comput. & Fluids 34 (2005), no. 6, 642–663. MR 2125442, DOI 10.1016/j.compfluid.2004.05.005
Xinghui Zhong and Chi-Wang Shu, Numerical resolution of discontinuous Galerkin methods for time dependent wave equations, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 41-44, 2814–2827. MR 2824155, DOI 10.1016/j.cma.2011.05.010