Analysis of Discontinuous Galerkin Methods with Upwind-Biased Fluxes for One Dimensional Linear Hyperbolic Equations with Degenerate Variable Coefficients

In this paper, we analyze the discontinuous Galerkin method with upwind-biased numerical fluxes for one dimensional linear hyperbolic equations with degenerate variable coefficients. The L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-stability is obtained by the choice of upwind-biased fluxes which could provide more flexible numerical viscosity. Furthermore, we construct some new piecewise global projections and present proofs of unique existence and optimal approximation properties. Then the optimal error estimates are derived by the benefits of the specially designed projections, essentially following the energy analysis. Numerical experiments are given which confirm the sharpness of the theoretical results.