Journal of Computational Physics
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Superconvergence and time evolution of discontinuous Galerkin finite element solutions
Journal of Computational Physics
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In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be $k+\frac{3}{2}$ when piecewise $P^k$ polynomials with $k\geq1$ are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise $P^k$ polynomials with arbitrary $k\geq1$, improving upon the results in [Y. Cheng and C.-W. Shu, J. Comput. Phys., 227 (2008), pp. 9612-9627], [Y. Cheng and C.-W. Shu, Computers and Structures, 87 (2009), pp. 630-641] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise $P^1$ polynomials.