The numerical methods that have been widely used for the solution of partial differential equations (PDEs), both in fluid dynamics and in other disciplines, fall into three main branches: finite difference methods, finite element methods, and finite volume methods. These methides are reviewed in this chapter together with basic theory of spectral methods.

This chapter examines the stability of difference schemes for initial value problems defined by ordinary or partial differential equations. Three simple examples are examined first: an ordinary differential equation, the linear advection equation, which is the prototype for hyperbolic equations, and the diffusion equations.