We have used a lower case **u** to denote the continuous field variable,
.
Note that all of the quadrature and difference formulas involving **u** are
stated as approximate equalities.
In each of these approximate equality statements, the amount by which
the right side differs from the left side is called the truncation error.
If **u** is a well behaved
function (has enough smooth derivatives), then it can be shown that these truncation
errors approach zero as the grid spacings, **h** and **k**, approach zero.

When we solve a finite difference equation, we obtain approximations to the values of the PDE problem solution at the grid points; that is, approximations to . To emphasize this approximation we write

where denotes the exact solution of a FDE problem on the grid
.
With this
notation in place, we are in a position to state three commonly used finite
difference
methods for dealing with a conservation law system in which **u** is the field
variable and density and flux are given by

and

Combining Eq. 12 with the difference quotient formulas of Eq. 19 and Eq. 20 we obtain the following.