Numerical Approximation of Surface and Volume Integrals

In Figs. 12.1 and 12.2, typical 2D and 3D Cartesian cell volumes are shown together with the notation commonly used. In 3D the GCV surface is subdivided into six plane surfaces, denoted by lower case letters corresponding to their direction e, w, n, s, t, and b) with respect to the central node P. The 2D case can be regarded as a special case of the 3D one in which the dependent variables are independent of z, hence the GCV surface is subdivided into four plane surfaces e, w, n, and s). 

The net flux through the GCV boundary is the sum of integrals over the GCV faces  

Many quadrature formulas approximate the integral by a weighted sum of the values of the integrand at particular points on the interval of integration, that is, by [57, 107, 155, 182]  

Several terms in the transport equations require integration over the volume of a grid cell. The midpoint rule is again the simplest second order approximation available. The second-order approximation thus consists in replacing the volume integral by the product of the mean value and the GCV. The mean value is approximated as the value at the GCV center  

Higher order approximations are possible but require the value of s at more locations than just the GCV center. If the intervals between the interpolation points are allowed to vary, other groups of quadrature formulas can be used, such as the Gaussian-, Clenshaw-Curtis- and FeJ6r quadrature formulas. However, in engineering practice the simple second order midpoint approximation formulas are normally used for the surface and volume integrals in the FVM. 

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