Finite Difference Methods for Hyperbolic Equations

Abstract

Solution of hyperbolic equations is perhaps the area in which finite difference methods have most successfully continued to play an important role. This is particularly true for nonlinear conservation laws, which, however, are beyond the scope of this elementary presentation. Here we begin in Sect. 12.1 with the pure initial-value problem for a first order scalar equation in one space variable and study stability and error estimates for the basic upwind scheme, the Friedrichs scheme, and the Lax-Wendroff scheme. In Sect. 12.2 we extend these considerations to symmetric hyperbolic systems and also to higher space dimension, and in Sect. 12.3 we treat the Wendroff box scheme for a mixed initial-boundary value problem in one space dimension.