Using Crank-Nicolson Method to solve Heat Equation

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Main Question or Discussion Point

I'm a bit stuck with using the C-N method

The question I'm trying to solve is the standard heat equation with:

U=[sin(pi)*x] at [itex]\tau[/itex] = 0

& U = 0 at x = 0

& x = 1 for [itex]\tau \geq 0[/itex]

The intervals are 0.2 in x AND 0.02 in [itex]\tau[/itex] up to [itex]\tau[/itex] = 0.06

I've been asked to solve using an Explicit which I've done using formula derived from the taylor theorem, but the second part is asking to use the C-N Method.

I started using this formula:
5e27aff609471f9ebbbce7b7dae13ee5.png
in matrix form, where I'm letting r = 0.5 using
b9735da76916e0f2514a600f1acd3dcd.png


The only thing that's concerning me is that it seems a bit long winded and the answers for U(i,j) that have arisen don't seem at all close to the explicit method.

I put it all in matrix form and then using Gaussian elimination, is the correct method?
 

Answers and Replies

  • #2
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Update 1

I've managed to work through the first matrix for u(n,1) and seem to have gotten a good set of results. What I've noticed is that the u(1,1) = u(4,1) & u(2,1) = u(3,1).

Is it worth just skipping out the 3rd and 4th row in the matrix as I can just input the results from u(1,n) and u(2,n) into u(3,n) and u(4,n)?

Update 2

I misread the question, which actually just asked for the first time step which I had already calculated.
 
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