Taylor series and finite differences

To formally analyse the error associated with a finite difference approximation for a derivative, one expands each possible term of the approximation in a Taylor series (TS) so that Dapprox = Taylor series. You then would take the difference between the approximation (LHS) and the derivative term from the TS (on RHS), and you are left with the 'error'. As h gets small, the biggest term of this series will dominate the error. You now have an explicit coefficient for E ~ O(hⁿ).

Replacing derivatives in differential equations with finite difference approximations, gives a system of algebraic equations that can be solved numerically. To analyse the error in this case, we now need a "local truncation error" at each grid point as well as some handle on the "global error" over the entire mesh.

Local truncation error