Superconvergence of Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Abstract

In this paper, we study the superconvergence properties of an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method with approximations to one-dimensional linear hyperbolic equations. The ALE-DG method is a mesh moving method; we need to deal with the new difficulties brought by the time-dependent test function space and grid velocity field. Since the time derivative cannot commute with the space projections for the ALE-DG method, we will introduce the material derivative in our analysis. With the help of the scaling argument and material derivative, we build a special interpolation function by constructing the correction functions and prove that the numerical solution is superclose to the interpolation function in the $L^2$-norm. The order of the superconvergence is 2k+1 when piecewise polynomials of degree at most $k$ are used. We also rigorously prove a (2k+1)th-order superconvergence rate for the domain and cell average and at the downwind points in the maximal and average norm. Furthermore, we prove that the function value approximation is superconvergent with a rate $k+2$ at the right Radau points and a superconvergence rate $k+1$ for the derivative approximation at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.