A beginner's course in finite volume approximation of scalar conservation laws
Title: A beginner's course in finite volume approximation of scalar conservation laws
Summary: in this lecture, we will present and study some methods to discretize a scalar conservation law $\partial_t u + \partial_x f(u)=0$. Considering first the case of a linear equation ($f(u)=au$) we will try and understand the basic construction of a numerical scheme (using Finite volume techniques), and the issues related, mainly concerning the stability of the method. We will then introduce the principle of monotone schemes for general non-linear equation, and give some classical examples (Lax-Friedrichs, Godounov); we will try to understand the concept of numerical diffusion associated with such fluxes (and its link with the discretization of parabolic equations), and we will give some elements of the study of such schemes: stability, discrete entropy inequalities, convergence in the BV case, convergence in the $L^\infty$ case. The numerical diffusion introduced by monotone fluxes allow to stabilize the scheme, but gives poor approximations of the shocks occuring in hyperbolic equations; in a last section, we will introduce some higher order methods (MUSCL techniques) which allow to obtain better qualitative approximation of the shocks.
Prerequisites: bases in mathematical analysis, knowledge of the properties of the solution to scalar conservation laws (shocks, rarefaction waves, Krushkov's entropy inequalities).
Some familiarity with BV spaces can help. No special knowledge of numerical analysis is required.
Jérôme DRONIOU
Professor of Mathematics, University of Montpellier II
dimanche 15 juin 2008
