In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A $(2k+1)$th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise polynomials of degree $k$ are used. Furthermore, surprisingly, we find that the derivative and function value approximation of the DG solution are superconvergent at a class of special points, with an order $k+1$ and $k+2$, respectively. These superconvergent points can be regarded as the generalized Radau points. All theoretical findings are confirmed by numerical experiments.

Keywords: Discontinuous Galerkin methods, superconvergence, generalized Radau points, upwind-biased fluxes

^{1}; Li, Dongfang

^{2}; Yang, Yang

^{3}; Zhang, Zhimin

^{1, 4}

@article{M2AN_2017__51_2_467_0, author = {Cao, Waixiang and Li, Dongfang and Yang, Yang and Zhang, Zhimin}, title = {Superconvergence of {Discontinuous} {Galerkin} methods based on upwind-biased fluxes for {1D} linear hyperbolic equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {467--486}, publisher = {EDP-Sciences}, volume = {51}, number = {2}, year = {2017}, doi = {10.1051/m2an/2016026}, mrnumber = {3626407}, zbl = {1367.65127}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016026/} }

TY - JOUR AU - Cao, Waixiang AU - Li, Dongfang AU - Yang, Yang AU - Zhang, Zhimin TI - Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 467 EP - 486 VL - 51 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016026/ DO - 10.1051/m2an/2016026 LA - en ID - M2AN_2017__51_2_467_0 ER -

%0 Journal Article %A Cao, Waixiang %A Li, Dongfang %A Yang, Yang %A Zhang, Zhimin %T Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 467-486 %V 51 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016026/ %R 10.1051/m2an/2016026 %G en %F M2AN_2017__51_2_467_0

Cao, Waixiang; Li, Dongfang; Yang, Yang; Zhang, Zhimin. Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 51 (2017) no. 2, pp. 467-486. doi : 10.1051/m2an/2016026. http://www.numdam.org/articles/10.1051/m2an/2016026/

Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3331–3346. | DOI | MR | Zbl

and ,Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3113–3129. | DOI | MR | Zbl

and ,Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems. Math. Comp. 80 (2011) 1335–1367. | DOI | MR | Zbl

and ,Superconvergence of discontinuous Galerkin method for 2-D hyperbolic equations. SIAM J. Numer. Anal. 53 (2015) 1651–1671. | DOI | MR | Zbl

, , and ,Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations. Math. Comp. 85 (2016) 63–84. | DOI | MR | Zbl

and ,Superconvergence of Discontinuous Galerkin method for linear hyperbolic equations.SIAM J. Numer. Anal. 52 (2014) 2555–2573. | DOI | MR | Zbl

, and ,Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys. 227 (2008) 9612–9627. | DOI | MR | Zbl

and ,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) 4044–4072. | DOI | MR | Zbl

and ,TVB Runge-Kutta local projection discontinuous Galerkin finite element method for coservation laws, II: Genaral framework. Math. Comput. 52 (1989) 411–435. | MR | Zbl

and ,The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, RAIRO: Model. Math Anal. Numer. 25 (1991) 337–361. | Numdam | MR | Zbl

and ,The Runge-Kutta discontinuous Galerkin method for conservation laws, V: Multidimensional systems. J. Comput. Phys. 141 (1998) 199–224. | DOI | MR | Zbl

and ,The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. | DOI | MR | Zbl

and ,The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, IV: The Multidimensional case. Math. Comp. 54 (1990) 545–581. | MR | Zbl

, 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. | DOI | MR | Zbl

, and ,Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. | DOI | MR | Zbl

, and ,Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: eigen-structure analysis based on Fourier approach. J. Comput. Phys. 235 (2013) 458–485. | DOI | MR | Zbl

, and ,Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput. 85 (2016) 1225–1261. | DOI | MR | Zbl

, and ,Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D. Math. Comput. 79 (2010) 35–45. | DOI | MR | Zbl

and ,Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50 (2012) 3110–3133. | DOI | MR | Zbl

and ,Superconvergence of discontinuous Galerkin methods for convection-diffusion problems. J. Sci. Comput. 41 (2009) 70–93. | DOI | MR | Zbl

, and ,*Cited by Sources: *