THE DISCONTINUOUS FINITE-ELEMENT METHOD WITH THE TAYLOR-GALERKIN APPROACH FOR NONLINEAR HYPERBOLIC CONSERVATION-LAWS

A one step, explicit finite element scheme which is second order accurate in both time and space is developed for the computation of weak solutions of nonlinear hyperbolic conservation laws in one dimension. The scheme is an improved version of the discontinuous finite element method using the Taylor-Galerkin procedure. It is linearly stable under a fixed CFL number up to nearly 0.4 (or 1/2 for the alternative two step explicit scheme also developed here) and TVDM which guarantees convergence to a weak solution in nonlinear problems when the flux limiter is applied with the CFL number up to 0.366 (or 1/2 for the two step scheme). Numerical experimentation demonstrates the convergence of the solution to the entropy one even for nonconvex flux cases. The scheme captures stationary discontinuities perfectly