Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

A Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Transient Linear Hyperbolic Conservation Laws on Cartesian Grids

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 [Formula: see text]-norm of the solution is of order [Formula: see text], when tensor product polynomials of degree at most [Formula: see text] are used. Then, we prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be [Formula: see text]. Moreover, we establish an [Formula: see text] global superconvergence for the solution flux at the outflow boundary of the domain. We also provide a simple derivative recovery formula which is [Formula: see text] 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 [Formula: see text]-norm at [Formula: see text] rate. Our results are valid without the flow condition restrictions. Numerical examples validating these theoretical results are presented.