With respect to dynamic meshes, the integral form of the conservation
equation for a general scalar, 
, on an arbitrary control volume, 
, whose boundary is moving
can be written as
 | (3–1) |
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where | |
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Here, 
is used to represent the
boundary of the control volume, 
.
By using a first-order backward difference formula, the time derivative term in Equation 3–1 can be written as
 | (3–2) |
where 
and 
denote the respective quantity
at the current and next time level, respectively. The 
th time
level volume, 
, is computed from
 | (3–3) |
where 
is the volume time derivative
of the control volume. In order to satisfy the mesh conservation law,
the volume time derivative of the control volume is computed from
 | (3–4) |
where 
is the number of faces on the
control volume and 
is the 
face area vector. The dot
product 
on each control volume face is calculated from
 | (3–5) |
where 
is the volume
swept out by the control volume face 
over the time step 
.
By using a second-order backward difference formula, the time derivative in Equation 3–1 can be written as
 | (3–6) |
where 
, 
, and 
denote the respective
quantities from successive time levels with 
denoting the current
time level.
In the case of a second-order difference scheme the volume time
derivative of the control volume is computed in the same manner as
in the first-order scheme as shown in Equation 3–4.
For the second-order differencing scheme, the dot product 
on each control volume face
is calculated from
 | (3–7) |
where 
and 
are the volumes swept out by control volume faces
at the current and previous time levels over a time step.