Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Conservation Laws in One Space Dimension

Abstract. In this paper, the analysis of the superconvergence property of the discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves (k+ 3 2 )-th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k (k ≥ 1), under the condition that |f (u)| possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on f(u) is artificial.