Finite Difference Methods and Global Error

O_chemist

I was going through my notes on different finite difference methods and came across something I don't quite understand. I have code that will calculate an approximate solution we can call this U_nm that I define on a grid using h and dt for the change in x and time respectively. Now I have written down the global error is just

e_nm =|U_nm - u(xn,tn)|

where u(xn,tn) is the exact solution evaluated at the gird points.

From there we can calculate our rate of convergence

So naively I just assumed I could take the solution I calculate and subtract the exact solution at every point take the absolute value. However, I have written something about actually just using e_nm to calculate the initial values as well as boundary conditions and then plugging them back into the finite difference method to calculate all the grid points for all of e_nm.

Is this correct or did I perhaps not fully understand what my instructor was saying?

(Note we are working with forward, backward and crank-nicolson methods)