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Math Help - finite difference method, coupled wave equations, chickens & eggs

  1. #1
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    Question finite difference method, coupled wave equations, chickens & eggs

    I'm reading a book (Numerical Techniques in Electromagnetics by Sadiku) & just finished the section on finite difference methods. As what I thought would be an easy exercise, I tried to apply what I'd learned to the telegraphers equations that describe the voltage, V(x, t), and current, I(x, t), on a transmission line with some realistic (meaning ugly) boundary conditions:

    finite difference method, coupled wave equations, chickens & eggs-tl.png

    The equations are 2 coupled (by the boundary conditions) wave equations:

    <br />
\frac{\partial^2 V}{\partial t^2} = u^2 \frac{\partial^2 V}{\partial x^2}<br />

    <br />
\frac{\partial^2 I}{\partial t^2} = u^2 \frac{\partial^2 I}{\partial x^2}<br />

    Where u is the velocity of propagation and 0 < x < L, t > 0
    The boundary & initial conditions are
    <br />
V(0, 0) = V_g(0) \frac{Z_0}{Z_0 + R_g}<br />

    <br />
I(0, 0) = V_g(0) \frac{1}{Z_0 + R_g}<br />

    <br />
V(0, t) = V_g(t) - R_g I(0, t)<br />

    <br />
V(L, t) = R_L I(L, t)<br />

    V(x, 0) = I(x, 0) = 0 for x>0 & everything(t<0) = 0
    RL, Rg, & Z0 are real positive constants. Vg(t) is a known function of time only.

    So I turn each into a difference equations using the centralized 2nd order approximation

    <br />
\frac{V(i, j+1) - 2 V(i, j) + V(i, j-1)}{\Delta t^2} = u^2 \frac{V(i+1, j) - 2 V(i, j) + V(i-1, j)}{\Delta x^2}<br />

    same for I. I solve for V(i, j+1) & I(i, j+1). With j (time) in my outer loop & i (x) in my inner loop I start stepping across x for each time t. Everything is fine until I reach that last boundary condition. I have a chicken & egg problem. First impulse is to step V forward and calculate I or vice versa. But neither will give me the right answer. I'm going to have the same problem at x=0 after that first step.
    The problem is easy if RL = Rg = 0. I can just solve the equation for V since V(L, t) = 0.

    But, for RL & Rg > 0, how do I handle those two boundary conditions that relate V & I at x=0 and x=L?

    I know there are other (& probably better) ways to solve this, even analytically or just intuitively. But I need the practice with FD.

    Thanks,
    Apchar
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  2. #2
    A Plied Mathematician
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    I'm no expert on FD, but it seems to me that your stepping approach is the problem. Typically, FD methods can handle boundary-value problems in a very straight-forward manner: use the discretized derivatives you have there to set up a large system of simultaneous equations. That's the key word: simultaneous. With your method, you're bootstrapping yourself up to get more and more values as you go along. Then, when you get to the end, you have a hard time matching up the final boundary conditions (although, if you have a fast enough computer, you could iterate through your scheme using a shooting method.) Think of a grid, x in one direction, and t in the other. The functions V and I are going to need to satisfy a certain system. You'll need to set up that system (the hard part), and then hand it over to a computer to solve.

    Like I said, I'm no expert, and we will very quickly exhaust my expertise in the subject. But maybe this will give you some ideas.
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  3. #3
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    The aforementioned text describes 2 FD methods. The implicit method as you described where, for each slice of time, you solve N simultaneous equations (where N is the number of spatial points) and the explicit method which is as I described. I was hoping to use the explicit method since it's so much simpler.
    What irks me most is that it works with RL = Rg = 0, RL = Rg = infinity, and RL = Rg = Z0. It just work in any other value of RL (& Rg.) The fact that it works for those conditions leads me to believe it can work for any RL (& Rg) but I can't for the life of me see how.
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  4. #4
    A Plied Mathematician
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    The explicit method may be simpler, but it's not suited to BVP's unless you couple it with a shooting method. That is, you end up having to solve an awful lot of IVP's to solve your BVP.
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