Numerical Methods for the solution of Hyperbolic Conservation Laws

In this paper we consider numerical approximations of hyperbolic conservation laws in the one-dimensional scalar case, by studying Godunov and van Leer's methods. Before to present the numerical treatment of hyperbolic conservation laws, a theoretical introduction is given together with the definition of the Riemann problem. Next the numerical schemes are discussed. We also present numerical experiments for the linear advection equation and Burgers' equation. The first equation is used for modeling discontinuities in fluid dynamics; the second one is used for modeling shocks and rarefaction waves. In this way we can compare the different behavior of both schemes