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. The scheme is an improved version of the discontinuous finite element (Discontinuous Galerkin) method using the Taylor-Galerkin procedure. It is linearly stable under a fixed CFL number up to nearly 0.4 (or {dollar}1over2{dollar} for the alternative two step explicit scheme also developed here) and TVDM which guarantees convergence to a weak solution in non-linear problems when the flux limiter is applied with the CFL number up to 0.366 (or {dollar}1over2{dollar} for the two step scheme) in one dimension. The scheme is easily extended to multi-dimensions by approximating the two-dimensional Riemann problem and using the extended flux limiter. No special treatment is required for the refined elements when using an adaptive finite element method. Numerical experimentation demonstrates the overall improvement in solution quality and the convergence of the solution to the entropy one even for non-convex flux cases. The scheme captures stationary dicontinuities perfectly.