This paper is concerned with the application of the discontinuous Galerkin (DG) method to the solution of unsteady linear hyperbolic conservation laws on Cartesian grids. We present several superconvergence results and we construct a robust recovery-type a posteriori error estimator for the directional derivative approximation based on an enhanced recovery technique. We first identify a special numerical flux and a suitable initial discretization for which the L2-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. Then, we prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. Moreover, we establish an (h2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. We also provide a simple derivative recovery formula which is (hp+1) superconvergent approximation to the directional derivative. We use the superconvergence results to construct asymptotically exact a posteriori error estimate for the directional derivative approximation by solving a local steady problem on each element. Finally, we prove that the a posteriori DG error estimate at a fixed time converges to the true error in the L2-norm at (hp+1) rate. Our results are valid without the flow condition restrictions. Numerical examples validating these theoretical results are presented.
- Discontinuous Galerkin method
- a posteriori error estimates
- derivative recovery technique
- hyperbolic problems
ASJC Scopus subject areas
- Computer Science (miscellaneous)
- Computational Mathematics