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In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al.[To appear in Math. Comput.DOI: https://doi.org/10.1090/mcom/3562(2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k+ min(3, k) when piecewise ℙkpolynomials with k≥ 2 are used. We also prove a 2k-th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of (k+ 2)-th and (k+ 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k≥ 2. Numerical experiments verify that all theoretical findings are sharp.

In this paper we present an a priori error estimate of the Runge–Kutta discontinuous Galerkin method for solving symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge–Kutta method and the finite element space is made up of piecewise polynomials of degree k≥ 2. Quasi-optimal error estimate is obtained by energy techniques, for the so-called generalized E-fluxes under the standard temporal-spatial CFL condition τ≤ γh, where his the element length and τis time step, and γis a positive constant independent of hand τ. Optimal estimates are also considered when the upwind numerical flux is used.