Let us consider (for simplicity in one dimension) the general local representation:
where 
is the average value, defined by:
In order to define completely the representation, the derivatives
and 
have to be estimated. If we require
these gradients to depend only on quantities of adjacent cells, the
only choice is to use
only central differences of averaged values, that is:
Setting 
and
within cells 
and 
gives the
interface values:
The TVD condition for non-linear stability can be easily obtained by the application of a limiter on the differences in the above formulae.