Métodos numéricos não oscilatórios aplicados às leis de conservação hiperbólicas unidimensionais

Abstract

The solution of a conservation law may develop discontinuities like shocks and rarefactions waves, even if the initial condition is a smooth function. Then, numerical schemes should be able to generate ecient approximations in order to reproduce the same behavior as the analytic solution. Besides, these schemes have to capture the physically correct solution or entropy solution. The goal of this master dissertation is to study non-oscillatory schemes applied to one-dimensional scalar hyperbolic conservation laws. In order to reach the is objective, it is necessary to understand some special methods, such as, upwind scheme, TVD schemes, conservative schemes and monotone schemes. The eectiveness of the methods will be veried through the comparison with the well-known classical solutions exhibited in literature: Advection Equation and Burgers' Equation. The characteristic equations will be employed for getting analytic solutions of conservation laws. We will derive numerical approximations for conservation laws using ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for space discretization, and Runge-Kutta TVD (Total Variation Dimishing) for time discretization.