Point-wise and cell average error estimates of the DG and LDG methods for 1D hyperbolic and parabolic equations

Abstract

In this paper, we study superconvergence properties of the discontinuous and local discontinuous Galerkin methods for 1D hyperbolic conservation laws and parabolic equations. Speci cally, when using upwind fluxes (for hyperbolic conservation laws) and alternating fluxes (for parabolic equations), we prove for any poly- nomial degree k , a (2 k + 1)-th superconvergence rate for the error of cell average, as well as the point-wise error estimates of the DG approximation at downwind points (hyperbolic) or the LDG approximation of numerical traces at nodes (parabolic). These superconvergence results are completely consistent with the numerical results provided in Cao et al. (2014), Cao and Zhang (2014) and Yang and Shu (2012), and improve the theoretical results in Cao et al. (2014) and Cao and Zhang (2014), which is 1/2 order and one order higher than that in Cao et al. (2014) and Cao and Zhang (2014) for the point-wise error estimates and cell average, respectively.