Conservative Property

A finite-difference equation posseses conservative property if it preserves integral conservation relations of the continuum. Let us consider the vorticity transport equation

(9.13)

where is nabla or differential operator,
         V the fluid velocity and
          the vorticity.

If we integrate this over some fixed space region we get

(9.14)

The first term of the Eq. (9.14) can be written as

The second term of the Eq (9.14) may be expressed as

is the boundary of ,
n is unit normal vector and
dA is the differential element of .

The remaining term of Eq. (9.14) may be written as

As such

Finally, we can write

(9.15)

which signifies that the time rate of accumulation of in is equal to net advective flux rate of across into plus net diffusive flux rate of across into .

The concept of conservative property is to maintain this integral relation in finite difference representation.

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