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Math Help - How to compute this weird derivative?

  1. #1
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    How to compute this simple but weird derivative?

    Hello,

    I am working on numerical solution of incompressible fluid dynamics problem. I need to compute the Jacobian of a viscous flux. I have to deduct some derivatives like this

    \frac{\partial (\frac{\partial U}{\partial X})}{\partial U}=?

    U, X are independent to each other.

    Some people say the result is absolutely not zero, but I still could not find an analytical answer. I wonder this will turn out into a derivative operator rather than an expression of fixed value. If an analytical expression is not possible, how to handle this in a finite difference method?


    Thank you and have a good day!

    Bearcat
    Last edited by bearcat; May 10th 2010 at 06:09 PM.
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  2. #2
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    Quote Originally Posted by bearcat View Post
    Hello,

    I am working on numerical solution of incompressible fluid dynamics problem. I need to compute the Jacobian of a viscous flux. I have to deduct some derivatives like this

    \frac{\partial (\frac{\partial U}{\partial x})}{\partial U}=?



    Some people say the result is absolutely not zero, but I still could not find an analytical answer. I wonder this will turn out into a derivative operator rather than an expression of fixed value. If an analytical expression is not possible, how to handle this in a finite difference method?


    Thank you and have a good day!

    Bearcat
    I've seen something like this before...

    What you want to do is:

    (1)Take the partial of U w.r.t. x. Denote the result by L.

    (2)Transform L so that it is in terms of U. (This is the hard part)

    (3)Take the partial of Transform(L) w.r.t. U.

    Can you post what U is?
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  3. #3
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    sorry, I need to add more info

    In my case, U is the unknown velocity in N-S equation, x is the coordinate. they are independent to each other. What I am looking for is an analytical expression for my coding. I am sure it's solvable in some way as I am not the first to encounter this problem.
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  4. #4
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    I suspect it's possibly   \frac{1}{\delta _{x}}  .
    Last edited by bearcat; May 11th 2010 at 07:35 AM.
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