TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems

This is the third paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut+Σi=1d(fi(u)xi=0. In this paper we present the method in a system of equations, stressing the point of how to use the weak form in the component spaces, but to use the local projection limiting in the characteristic fields, and how to implement boundary conditions. A 1-dimensional system is thus chosen as a model. Different implementation techniques are discussed, theories analogous to scalar cases are proven for linear systems, and numerical results are given illustrating the method on nonlinear systems. Discussions of handling complicated geometries via adaptive triangle elements will appear in future papers.