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Stability condition for solving convection equation by FDM

  1. May 25, 2015 #1
    Hi,

    I know, there is a stability condition for solving the Convection-Diffusion equation by Finite Difference explicit/implicit technique, which is \Delta t<=(\Delta x)^2/(2*D) for one-dimensional or \Delta t<=((\Delta x)^2+(\Delta y)^2)/(8*D) for two-dimensional problem, where D is the diffusion coefficient.
    .
    Is there any such condition for only the convection equation?

    Thanks
     
  2. jcsd
  3. May 27, 2015 #2
    The stability criterion is dependent on your FD approximation of the laplacian. Have you tried calculating the stability conditions? I am not sure why the same method wouldn't work for the convection term. I have not done it for the convection though.
     
  4. May 27, 2015 #3
    Thanks for your response. I am not sure how to calculate the stability condition. I have used the the formula \Delta t<=((\Delta x)^2+(\Delta y)^2)/(8*D) to make my program stable. But if there is no diffusion term,only convection term, how will I calculate the stability condition?

    Thanks
     
  5. May 28, 2015 #4
    After some thought I am not even sure you can do the calculation with the convection term, since the stability of the diffusion equation is calculated using a fourier series. Unless the velocity is constant i suppose things will not work that well.
    Further more as far as I remember just advecting a field is not so easy numerically due to various problems(at least not in the long time limit). What are you advecting by the way? An interface? This article discusses a bit about passiv advection
    http://www.ias.ac.in/sadhana/Pdf2009Apr/271.pdf

    I think it is nice enough.(and eq 2 might be what you are looking for).
     
  6. May 28, 2015 #5
    Thanks.
     
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