Computational methods for hyperbolic equations

This is an introduction to some of the basic concepts on modern numerical methods for computing approximate solutions to hyperbolic partial differential equations. This chapter is divided into five sections. Section 1 contains a review of some elementary theoretical concepts on hyperbolic equations, mainly focused on the linear case; the Riemann problem for a general linear system with constant coefficients is solved in detail. Section 2 is an introduction to the basics of discretization methods, including finite difference methods and finite volume methods; concepts such as local truncation error, linear stability and modified equation are included; Godunov's theorem is stated, proved and its implications are discussed. Section 3 contains two approximate Riemann solvers, as applied to the three-dimensional Euler equations, namely HLLC and EVILIN. Section 4 deals with the construction of non-linear (non-oscillatory) numerical methods of the TVD and ENO type, for a scalar conservation law. In Sect. 5 we use the theory developed for scalar equations as a guideline to construct non-linear (quasi non-oscillatory) second-order finite volume schemes for one-dimensional non-linear systems with source terms. Key references for further reading are indicated at the end of each section. © 2008 Springer-Verlag Berlin Heidelberg.