An approximate evolution of Eqn. (15.5) would be

(15.7)

where A is the area of the quadrilateral ABCD in Fig. 15.1, and the average value of E over the quadrilateral is represented by and the remaining terms are approximations for the line integral over segments and respectively.

Further, Eqn. (15.8) follows,

(15.8)

For the irregular grid-mesh the finite volume Eqn. (15.8) provides a discretisation in Cartesian coordinates without introducing generalized coordinates. If the grid-mesh is uniform and coincides with line of constant and Eqn. (15.8) becomes

or,

(15.9)

Which coincides with a central difference representation for the spatial terms of (15.1). The finite volume method, which is extensively used for both incompressible and compressible flows, has the advantages of conservative property.

Most importantly, it allows complex computational domains to be discretised in a simple way.

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