# Exact value of truncation error

1. Sep 6, 2011

### roldy

How would you calculate the exact value of the truncation error? This is of course for finite element analysis using the forward finite difference method.

If your given a function u=u(x,t) and are to find the error at node (i,n+1), wouldn't you just take the difference between the value of the function and the value of the finite method?

So for example
u=u(x,t)=x*sin(t)
du/dt=x*cos(t)

finite difference method-time derivative:
$\frac{\partial ^u}{\partial t}=\frac{u_i^{n+1}-u_i^n}{\Delta t}=\frac{x*sin(t+\Delta t)-x*sin(t)}{\Delta t}$

So therefore the exact error is...
$x*cos(t)-\left[\frac{x*sin(t+\Delta t)-x*sin(t)}{\Delta t}\right]$

However, if you were to plug in numbers for x, t, and $\Delta t$, I don't see a way of calculating an exact value for the truncation error. Is my thought process wrong?

2. Sep 6, 2011

### Dickfore

If you know the exact error, you can evaluate the exact value.

3. Sep 6, 2011

### roldy

Assuming that my error equation is correct, that should result in the error. One other thing that I'm curious about is the leading truncated term. I've seen this a couple times before but I can not find the site where it talks more in depth. What is the leading truncated term? The only thing I can think of is the first derivative term dropped from the finite difference expression. Say for example you only want the first three terms truncated. Then in your expression you would include the value of the function at t, 1st derivative term, and the 2nd derivative term. Therefore the 3rd derivative term would be the leading truncated term. Is this correct?