Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha-th order (1 <= \alpha <= k+1) divided difference of the DG error in the L2-norm is of order k+(3-alpha)/2 when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order k+(3-alpha)/2 in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least (3k/2+1)th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order k+1 in the L2-norm for the divided differences of DG errors and thus (2k+1)th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results

Fast X-Ray Fluorescence Microtomography of Hydrated Biological Samples

Metals and metalloids play a key role in plant and other biological systems as some of them are essential to living organisms and all can be toxic at high concentrations. It is therefore important to understand how they are accumulated, complexed and transported within plants. In situ imaging of metal distribution at physiological relevant concentrations in highly hydrated biological systems is technically challenging. In the case of roots, this is mainly due to the possibility of artifacts arising during sample preparation such as cross sectioning. Synchrotron x-ray fluorescence microtomography has been used to obtain virtual cross sections of elemental distributions. However, traditionally this technique requires long data acquisition times. This has prohibited its application to highly hydrated biological samples which suffer both radiation damage and dehydration during extended analysis. However, recent advances in fast detectors coupled with powerful data acquisition approaches and suitable sample preparation methods can circumvent this problem. We demonstrate the heightened potential of this technique by imaging the distribution of nickel and zinc in hydrated plant roots. Although 3D tomography was still impeded by radiation damage, we successfully collected 2D tomograms of hydrated plant roots exposed to environmentally relevant metal concentrations for short periods of time. To our knowledge, this is the first published example of the possibilities offered by a new generation of fast fluorescence detectors to investigate metal and metalloid distribution in radiation-sensitive, biological samples

Investigation on the Effects of Recycling and Injection Parameters on Gloss Properties of Smooth Polypropylene Parts

International audienc