Notions on Numerical Methods

This chapter is a succinct, largely self-contained presentation of some very basic notions on numerical methods for solving hyperbolic equations. The subject is dealt with entirely in terms of the simplest partial differential equation (PDE), namely the linear advection equation with constant wave propagation speed. We begin with the simplest numerical approximation method, namely the finite difference method. Most well-known schemes, as finite difference methods, are presented, including the Lax-Friedrichs method, the Lax-Wendroff method, the FORCE method, the Godunov centred method and the Godunov upwind method. The main properties of these schemes are also studied, including truncation error, accuracy, monotonicity and linear stability. Some theoretical notions are also included, notably the Godunov theorem, which is stated and proved. This result sets the theoretical bases for the construction of non-linear methods for hyperbolic equations. Sample numerical results are presented. Useful background is found in Chap.  2 .