$Why does this spatial discretization with n intervals have a position of (n-1)/n for each interval?$

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$Why does this spatial discretization with n intervals have a position of (n-1)/n for each interval?$

Hello, all.

I am learning about how to descritize the Heat Equation in space. This case involves a rod along the xi-axis with some thermal conductivity, kappa.

Now, I understand that one would descritize it so that they can utilize a set of points to sample and approximate the behavior of a system as a whole.

According to the following picture, you can descritize the rode into a finite number of elements, with the quantity of these being n. I believe that I understand why each interval has a length of 1/n. My brain is connecting to the the frequency and period of a wave, where f = 1/T. Let's say that I want to have 100 samples, then the spatial distance between each equidistant sample must be 1/100 so that x goes from [0,1].

However, I do not understand why the spatial variable can be represented as (n-1) / n, as in the following equation. Can someone explain why? Also, I notice that the equation at the bottom changes an "n" to an "i". Could this be a typo?

Discrete Mathematics Partial Differential Equations Heat Equation

Exactplace

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The $i$th subinterval is $[(i-1)/n,i/n]$. Now $T$ is computed at the left endpoint of each subinterval as a sample point from that subinterval. And there are $n$ left endpoints, and the left endpoint of the $n$th interval is $(n-1)/n$.

Also the $i$ in the last equation is used correctly. In order to approximate the second derivative we can use $$ \partial^2_\xi T(t,x)\approx \frac{1}{h^2}[T(t,x+h)+T(t,x-h)-2T(t,x)].$$ Now set $x=(i-1)/n$ and $h=1/n$ in the above formula to get the desired.

Martin

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Exactplace

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Thank you for your help, Martin. May I ask where the approximation for the 2nd-order PDE comes from? A link to a video or PDF would be great, if possible.
Martin

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You're welcome. You can check https://en.wikipedia.org/wiki/Symmetric_derivative#The_second_symmetric_derivative and its references. It can also be shown by expanding each term using 2nd order Taylor approximation, then the lower order terms cancel each other.

$Why does this spatial discretization with n intervals have a position of (n-1)/n for each interval?$

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