THE RUNGE-KUTTA LOCAL PROJECTION RHO-1-DISCONTINUOUS-GALERKIN FINITE-ELEMENT METHOD FOR SCALAR CONSERVATION-LAWS

In this work we introduce and analyze the model scheme of a new class of methods devised for numerically solving hyperbolic conservation laws. The construction of the scheme is based on a Discontinuous Galerkin finite element space-discretization, combined suitably with a high-order accurate total variation diminishing Runge-Kutta time-discretization, and a local projection which enforces the global stability of the scheme. The resulting scheme verifies a maximum principle, is total variation bounded in the means, linearly stable for CFL is-an-element-of [0, 1/3], and formally uniformly second-order accurate in time and space. Moreover, it converges to a weak solution. We give extensive numerical evidence that the scheme does converge to the entropy solution, and that the order of convergence away from singularities is optimal; i.e., equal to 2 in the norm of L infinity (L(loc)infinity).