Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper, we study superconvergence properties of the discontinuous Galerkin (DG) method for one-dimensional linear hyperbolic equations when upwind fluxes are used. We prove, for any polynomial degree $k$, the $2k+1$th (or $2k+1/2$th) superconvergence rate of the DG approximation at the downwind points and for the domain average under quasi-uniform meshes and some suitable initial discretization. Moreover, we prove that the derivative approximation of the DG solution is superconvergent with a rate $k+1$ at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.

[1]  Slimane Adjerid,et al.  Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem , 2006 .

[2]  T. Weinhart,et al.  Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems , 2009 .

[3]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[4]  Bernardo Cockburn,et al.  The Runge-Kutta local projection discontinous Galerkin finite element method for conservation laws , 1990 .

[5]  WAIXIANG CAO,et al.  Is 2k-Conjecture Valid for Finite Volume Methods? , 2014, SIAM J. Numer. Anal..

[6]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[7]  Slimane Adjerid,et al.  Discontinuous Galerkin error estimation for linear symmetrizable hyperbolic systems , 2011, Math. Comput..

[8]  Chi-Wang Shu,et al.  Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension , 2010, SIAM J. Numer. Anal..

[9]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[10]  Zhimin Zhang,et al.  Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D , 2010, Math. Comput..

[11]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[12]  Karen Dragon Devine,et al.  A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems , 2002 .

[13]  Chuanmiao Chen,et al.  The highest order superconvergence for bi-k degree rectangular elements at nodes: A proof of 2k-conjecture , 2012, Math. Comput..

[14]  Zhimin Zhang,et al.  Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems , 2009, J. Sci. Comput..

[15]  Chi-Wang Shu,et al.  Superconvergence and time evolution of discontinuous Galerkin finite element solutions , 2008, J. Comput. Phys..

[16]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[17]  Wei Guo,et al.  Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods: Eigen-structure analysis based on Fourier approach , 2013, J. Comput. Phys..

[18]  Chi-Wang Shu,et al.  Analysis of Optimal Superconvergence of Discontinuous Galerkin Method for Linear Hyperbolic Equations , 2012, SIAM J. Numer. Anal..

[19]  Zhimin Zhang.  Superconvergence Points of Spectral Interpolation , 2012, 1204.5813.

[20]  Bernardo Cockburn,et al.  The Runge-Kutta local projection P1-discontinuous-Galerkin finite element method for scalar conservation laws , 1988 .