Numerical Analysis Group Research Report NA-93/10

An upwind finite element method for nonlinear hyperbolic systems of conservation laws

P. Lin and K. W. Morton

This paper proposes a new finite element method for approximating the numerical solution of hyperbolic systems of conservation laws in one dimension where strict hyperbolicity is not required. The method uses a special continuous parameterization technique and the theory of the Riemann-Stieltjes integral, which are the basic techniques already used in the Euler Characteristic Galerkin (ECG) method designed successfully for scalar conservation laws. The proposed scheme is conservative. For linear hyperbolic systems of conservation laws, it is proved to be TVD for the characteristic variables and TVB for the conserved variables. For general nonlinear hyperbolic systems, we prove that under some appropriate conditions, the limit function of the numerical solutions is an admissible solution of the conservation laws, satisfying the standard entropy inequality. Large time steps and non-uniform meshes are the main features of this method. Some numerical experiments for large CFL numbers are tested.