How do we choose the number of subintervals?

[evinda]

Hello! (Wave)

I have written a code to approximate the solution of the heat equation. I want to consider uniform partitions in order to approximate the solution of the given boundary / initial value problem.

So we partition $[a,b]$ in $N_x$ subintervals with length $h=\frac{b-a}{N_x}$, where the points $x_i, i=1, \dots ,N_x+1$, are given by the formula $x_i=a+(i-1)h$, and so we have $a=x_1<x_2< \dots <x_{N_x}<x_{N_{x+1}}=b$ and respectively we partition $[0,T_f]$ in $N_t$ subintervals of length $\tau=\frac{T_f-t_0}{N_t}$ and the points are $t_n=t_0+(n-1)\tau, n=1, \dots ,N_t+1$, so we have $t_0=t_1<t_2<\dots< t_{N_t}<t_{N_{t+1}}=T_f$.


Is there a criterion to choose $N_x$ and $N_t$ ? (Thinking)

 

[Ackbach]

Well, if you don't have the exact solution available, then one way to show your partition is good enough is to pretend your metric space is complete (Cauchy sequences converge). What I mean by that is you would show that your partition is good enough if, say, you divided each subinterval in half, and your solution did not differ from your previous solution by more than some pre-specified tolerance. The hope is that you're working in a complete space, so that if your solutions differ from each other by a very little bit, they are close to the true solution. Does that make sense?

 

[evinda]

In our case, the solution is given.

 

[Ackbach]

In that case, you would compare your approximation with the exact solution, and see if it's in some tolerance. Now, you have to be careful how you compare. I would probably do something like this: pick a bunch of random points in the $(x,t)$ plane, evaluate your approximation there and the exact solution there, square the difference (or take its magnitude), and add them all up. You might have some predetermined method of comparison, but I would think it would need to be something like this method.