finite difference methods and global error

Question

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: $$e_{nm} =|U_{nm} - u(x_n,t_n)|$$ where $u(x_n,t_n)$ is the exact solution evaluated at the gird points. From there we can calculate our rate of convergence.

So I, perhaps naively, 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)

Answer 1

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 $e_{nm}$.

This statement doesn't make any sense, so you can safely assume you missed something in the lecture :) In the general case you will not have access to the true solution, your instructor may have described some strategies for evaluating the error or convergence in this situation.