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Crank-Nicholson method for solving telegraph quation

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Hello,
I'm trying to solve telegraph equation (transmission line) using Crank-Nicholson in MATLAB, but I'm stuck with Crank-Nicholson difference scheme. Can someone help?
I'm using simplified model
  6 Comments
Torsten
Torsten on 31 Aug 2023
Edited: Torsten on 31 Aug 2023
You can try this, but it's not Crank-Nicolson. I'd call it Explicit Euler.
HD
HD on 1 Sep 2023
So can i write it like this
My question now is can I substitute in (2) using (1) and how can I write ?
If i use this eq , and if i write what is correct way to write ? Thanks

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Answers (1)

Torsten
Torsten on 1 Sep 2023
Moved: Torsten on 1 Sep 2023
It's unusual that the lower index denotes the time discretization - thus I will write y_{i}^{n} for y at time t(n) in grid point x(i).
Your CN discretization reads
(u_{i}^{n+1}-u_{i}^{n})/dt = (v_{i}^(n+1)+v_{i}^{n})/2
(v_{i}^{n+1}-v_{i}^{n})/dt = 1/(LC) * (u_{i+1}^{n+1}-2*u_{i}^{n+1}+u_{i-1}^{n+1} + u_{i+1}^{n}-2*u_{i}^{n}+u_{i-1}^{n})/(2*dx^2)
or
u_{i}^{n+1}/dt - v_{i}^(n+1)/2 = u_{i}^{n}/dt +v_{i}^{n}/2
-1/(LC)*(u_{i-1}^{n+1}-2*u_{i}^{n+1}+u_{i+1}^{n+1})/(2*dx^2) + v_{i}^{n+1}/dt = v_{i}^{n}/dt + 1/LC*(u_{i+1}^{n}-2*u_{i}^{n}+u_{i-1}^{n})/(2*dx^2)
This is a system of linear equations in the unknowns u_{i}^{n+1} and v_{i}^{n+1} (2<=i<=nx-1).
For indices i = 1 and i = nx, you have to incorporate the spatial boundary conditions for u and v = du/dt.
  25 Comments
Torsten
Torsten on 23 Sep 2023
Edited: Torsten on 23 Sep 2023
I'm not sure whether this averaging 1/2*(unew + uold) in Crank-Nicolson also applies to algebraic equations like the boundary conditions.
You might want to compare the results to simply setting
u_5^(n+1) = f(t^(n+1))
v_5^(n+1) = f'(t^(n+1))
thus setting the (2x2) matrix in A to [1 0;0 1], in B to [0 0 ; 0 0] and the vector b to
b = [0; 0; 0; 0; 0; ... ; f(t^(n+1)) ;f'(t^(n+1)) ]

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