Notions on Hyperbolic Equations

This Chapter is a succinct introduction to basic notions on the theory of hyperbolic equations. The linear advection equation and general linear hyperbolic systems are studied in some detail, including the concepts of characteristics, eigenstructure, hyperbolicity and the Riemann problem. For the non-linear case we focus on scalar equations; the concepts of integral forms, shock formation, Rankine-Hugoniot condition, non-uniqueness of discontinuous solution and the Lax entropy condition are introduced. The complete solution of the Riemann problem for Burgers’ equation is given. The material is tailored to the aims of this book; it furnishes the bases for analysing the mathematical and physical character of the non-linear shallow water equations in Chaps.  4 and 3 ; and for solving the Riemann problem in Chaps.  6 and 7 . The contents are also useful for designing and interpreting numerical methods for wave propagation phenomena.