Superconvergence of local discontinuous Galerkin methods for one-dimensional convection-diffusion equations

  • Authors:
  • Yingda Cheng;Chi-Wang Shu

  • Affiliations:
  • Department of Mathematics and ICES, University of Texas, Austin, TX 78712, United States;Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

  • Venue:
  • Computers and Structures
  • Year:
  • 2009

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Abstract

In this paper, we study the convergence behavior of the local discontinuous Galerkin (LDG) methods when applied to one-dimensional time dependent convection-diffusion equations. We show that the LDG solution will be superconvergent towards a particular projection of the exact solution, if this projection is carefully chosen based on the convection and diffusion fluxes. The order is observed to be at least k+2 when piecewise P^k polynomials are used. Moreover, the numerical traces for the solution are also superconvergent, sometimes, of higher-order. This is a continuation of our previous work [Cheng Y, Shu C-W. Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J Comput Phys 2008;227:9612-27], in which superconvergence of DG schemes for convection equations is discussed.