Numerical Discretization of Coupling Conditions by High-Order Schemes

We consider numerical schemes for 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} hyperbolic conservation laws on graphs. The hyperbolic equations are given on arcs which are one-dimensional in space and are coupled at a single point, the node, by a nonlinear coupling condition. We develop high-order finite volume discretizations for the coupled problem. The reconstruction of the fluxes at the node is obtained using derivatives of the parameterized algebraic conditions imposed at the nodal points in the network. Numerical results illustrate the expected theoretical behavior.