The reconstruction via primitive variables provides a very elegant solution
to the problem of approximating nodal values of a function once given its
cell-averages. This approach was apparently first introduced by Colella and
Woodward (1984)
[2] in their Parabolic-Piecewise Method (PPM), and
reproposed in the framework of ENO schemes by Harten et al. (1986),
[10].
The basic idea is quite simple: given a set of cell-averaged
values 
, the fundamental theorem of calculus allows to
compute the exact values of the primitive function at the cell-interfaces
by an arbitrary additive constant. The primitives values can be now
interpolated in some way, providing a polynomial reconstruction of the
primitive function, and the derivative of this polynomial gives the
desidered nodal reconstruction.
Let us formalize the procedure: assume that 
be a set of
cell-averages of a piecewise-smooth function 
, which is the derivative
of another (smooth) function 
Eq. (38) is equivalent to
The discrete pointwise-values at any cell-interface position
can be computed by applying the
following relation:
The lower limit 
is arbitrarly fixed and any point could
actually be chosen. ENO approximations are
not affected by this choice, because a change in the lower limit only
shifts by a constant value, and the property defined by
Eq. (38) which the ENO algorithm effectively uses is
unaffected by a constant shift. If we assume that we are starting with
the exact cell-averages 
, this procedure will give us the exact
pointwise values of the primitive functions 
at the cell-interfaces
. We can apply the interpolation to the ``smoothest''
collection of discrete values of the primitive function 
and
obtain an ENO interpolating polynomial:
We can then obtain an approximation to which we
call 
, by differentiating 
:
Since the primitive function is by one derivative smoother
than , a 
-degree polynomial interpolation provides a 
-order
accurate approximation, where 
. Hence, we lose one order of accuracy
because of the differentiation and we obtain:
