Numerical solutions of problems in fluid dynamics usually are formulated using one of two methods: the finite-volume method and the finite-difference method[4]. In the finite-difference approach, a finite-difference approximation of the differential equation is solved. The equation is first transformed from the physical domain to a uniform computational domain, and the differential form of the equation is usually solved at the node points. By contrast, the finite volume approach solves the integral form of the equation. The main advantage of the finite-volume approach over the finite-difference approach is that it is more suitable for complex geometry because the equation can be discritized directly in the ``physical space.''
The convection equation given in equation 2 is in the differential form. By applying Green's Thereom, the integral form of the convection equation takes on the form [4]:
where it is assumed that the velocity field satisfies the divergence-free
condition. In equation 3, denotes the cell area, and the flux
functions
and
are defined as
In equation 3, the function is defined at the cell-centered location.
In the cell-centered scheme, the dependent variables are defined at the
center of the cell. Rewrite equation 3 as:
Where the residual is defined as: