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