Lectures on hyperbolic equations and their numerical approximation

These introductory lecture notes on numerical methods for hyperbolic equations are suitable for advanced undergraduate and postgraduate students in mathematics and engineering disciplines. More advanced approaches exist and will be indicated as appropriate. The material is divided into four sections. Section 1 presents an overview of hyperbolic equations and also some basic concepts on numerical discretization techniques. Section 2 deals with a specific example, the system of non-linear shallow water equations; the equations are analysed and the Riemann problem is solved exactly in complete detail. In Sect. 3 we first present the Godunov method as applied to a generic hyperbolic system and then specialised to the shallow water system in one space dimension; approximate solution methods for the Riemann problem are also given. Finally, Sect. 4 gives a brief overview of the ADER approach to construct high-order numerical methods for hyperbolic equations, based on the first order Godunov method. Much of the material of these lectures has been taken from the author’s text books (Toro, Riemann solvers and numerical methods for fluid dynamics. A practical introduction, 3rd edn. Springer, Berlin (2009) and Toro, Shock-capturing methods for free-surface shallow flows. Wiley, Chichester (2001)), where further reading material can be found. I also recommend the textbook by Godlewski and Raviart (Numerical approximation of hyperbolic systems of conservation laws. Springer, New York (1996)) and that by LeVeque (Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2002)).