ABSTRACT
In this chapter, basic concepts and techniques needed in the formulation of ƒnite-difference and ƒnite-volume representations are developed. The two formulations are closely related. Recall that the most fundamental statement of conservation principles such as conservation of mass, energy, or momentum applies to a ƒxed quantity of matter. From that starting point it is possible to develop conservation statements applicable to both a ƒxed region in space and at a point (in the limit of a vanishing volume). The conservation statement applicable to a point appears as a partial differential equation (PDE) and the statement for a ƒxed region in space as an equation involving integrals. In the ƒnite-difference formulation, the continuous problem domain is “discretized,” so that the dependent variables are considered to exist only at discrete points. The PDE form of the conservation statement is converted to an algebraic equation by approximating derivatives as differences. In the ƒnite-volume methodology, the continuous problem domain is divided up into ƒxed regions called control volumes. The dependent variables are considered to exist at a speciƒed location within the volumes or on the boundaries of the volumes. The integrals in the conservation statement for ƒxed regions in space are approximated algebraically. Thus, in both methods, a problem involving calculus has been transformed into an algebraic problem.
