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)

 

[gtoguy97]

It is correct that you can use e_nm to calculate the initial values and boundary conditions for your finite difference method. This allows you to achieve a more accurate solution by starting with values that are closer to the exact solution. However, this is not necessary for all finite difference methods. Generally speaking, this technique is used when the computed solution is inaccurate due to numerical errors or ill-conditioned systems.