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Finite Difference Method, Leapfrog (2,4) CFL Condition

  1. Dec 30, 2012 #1
    Hi.

    I'm trying to determine the CFL condition for the fourth-order leapfrog scheme. I'm finding 2 as what's published, which does not match what I'm getting.

    Does anyone know where I can find a von Neumann (or Fourier) stability analysis of the leapfrog (2,4) scheme (so I can compare my work) and/or a reputable book dedicated towards von Neumann (or Fourier) stability analysis?

    Thank you for your time,
    Will
     
    Last edited: Dec 30, 2012
  2. jcsd
  3. Dec 30, 2012 #2

    AlephZero

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    When you say "your calculations", do you mean numerical experiments with a finite number of mesh points and some specific boundary conditions? Or do you mean your own attempt at a theoretical stability analysis?

    If you mean numerical experiments, I would not be surprised that 2 is the limit for an infinitely large mesh. Try mesh sizes of n, 2n, 4n, 8n... for some reasonable value of n, and see what happens.
     
  4. Dec 30, 2012 #3
    Hi AlephZero.

    Thank you for the reply. No, I'm not referring to numerical experiments. I'm referring to the theoretical CFL condition and to a very basic application of von Neumann stability analysis where u^m_n = (g^m)(exp(i*xi_n*h)) and the scheme is u^(m+1)_n = u^(m-1)_n + lambda((4/3)(u^m_(n+1) - u^m_(n-1)) - (1/6)(u^m_(n+2) - u^m_(n-2)). I would prefer to determine the theoretical CFL condition, so I know what step sizes to gather my data in for preliminary confirmation of fitting the predicted with the observed and then subsequent experiments. I could just accept what's already been published, but I want to make sure I have a solid understanding of what's in front of me before I progress.

    Are you implying the published CFL condition is referring to actual numerical experiments or is it theoretical?

    Thank you for your time,
    Will
     
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