Due to the increasing complexities encountered in the development of modern technology, analytical solutions usually are not available. For these problems, numerical solutions obtained using high-speed computer are very useful, especially when the geometry of the object of interest is irregular, or the boundary conditions are nonlinear. In numerical analysis, three different approaches are commonly used: the finite difference, the finite volume and the finite element methods. In heat transfer problems, the finite difference and finite volume methods are used more often because of its simplicity in implementation.

Finite difference method involves:

Establish nodal networks:

The basic idea is to subdivide the area of interest into sub-volumes with the distance between adjacent nodes by $\Delta$x and $\Delta$y as shown. If the distance between points is small enough, the differential equation can be approximated locally by a set of finite difference equations. Each node now represents a small region where the nodal temperature is a measure of the average temperature of the region

i) Derive finite difference approximations for the governing equation at both interior and exterior nodal points

ii) Develop a system of simultaneous algebraic nodal equations

iii) Solve the system of equation using numerical schemes.