Time Scale Calculus, Discrete Exterior Calculus, Navier-Stokes

Discussion in 'Math Research' started by rfamgm, Apr 14, 2008.

  1. rfamgm

    rfamgm Guest

    This post is made to ask for comments about relationship between time
    scales and exterior derivative.

    First of all, I willl describe the situation:

    Taylor's Formula has a discrete analog in Newton's Forward Difference
    Formula.
    This and other analogs of continuous identities are derived in Umbral
    calculus.

    An alternative approach to working with discrete/continuous analogs
    was to unify them. This was done by Stephan Hilger who developed a
    generalized derivative on a measure chain (or time scale) which
    unified the study of difference equations and differential equations
    leading to dynamic equations on time scales.

    Martin Bohner and Gusein Guseinov have extended the study of dynamic
    equations on time scales to a multivariable calculus leading to
    partial dynamic equations on time scales which unifies partial
    difference equations with partial differential equations.
    M. Bohner has also developed a divergence, gradient and laplacian.

    In differential Geometry, the vector analysis operators are seen as 3D
    cases of the n-dimensional exterior derivative and in Anil Hirani's
    PhD thesis, a discrete exterior calculus is developed including
    discrete versions of Div, Grad, Curl and Lapacian.

    Now, my question is this: Do the definitions of Bohner's Time Scale
    vector operators and Hirani's Discrete exterior vector operators
    coincide. If not, why not ? Or if so, can time scales be used to unify
    discrete exterior calculus with standard exterior calculus ?

    Secondly, how are these definitions related to discrete versions of
    the Navier-Stokes equations.
    And if the discrete version can be solved, can time scale calculus be
    used to go from there to a solution of the continuous version.

    P.S. Can time scales be combined with p-adic numbers in any useful
    way ?

    Anil Hirani's PhD thesis:
    http://etd.caltech.edu/etd/available/etd-05202003-095403/unrestricted/thesis_hirani.pdf

    M.Bohner's publications:
    http://web.mst.edu/~bohner/pub.html
     
    rfamgm, Apr 14, 2008
    #1
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  2. Related to this is some work I did on "space time circuits":

    http://www.xs4all.nl/~westy31/Electric.html#SpaceTime

    The references you mention seem to talk about the same thing, but in a
    more formal language.

    [..]
    Well, in a space time circuit, the derivatives in the time direction are
    treated the same as in space directions, but with a negative impedance.
    Also, there is a clear relationship between exterior derivatives and the
    "coboundary operator" on the circuit.
    I also did the Navier Stokes:
    http://www.xs4all.nl/~westy31/Electric.html#Navier-Stokes

    The discrete Navier Stokes as I presented it, appears to be quite well
    behaved in a numerical simulation: I even did a 4-dimensional case! (I
    will put it on the web next week). The problem with "DNS" (Direct
    Numerical Simulation) of the Navier Stokes, is that you need a huge
    (increasing with Reynolds number) amount of cells to simulate the
    smallest flow structures. By using a much courser discretization, you
    get a solution, but you underestimate turbulence. But if you use
    sufficient cells, I believe the solution will be correct. But to *prove*
    that, you would need to solve one of the 7 Millennium problems!

    Gerard
     
    Gerard Westendorp, Apr 27, 2008
    #2
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  3. rfamgm

    caspro Guest

    Do you think that your electrical circuit diagrams/discrete exterior
    calculus can be used to prove the Four color theorem in a more
    illuminating way than Appel and Haken's computer-assisted proof ?
     
    caspro, May 15, 2008
    #3
  4. Well, I don't know much about the 4 color theorem.
    But you might relate it to the theory of circuit diagrams, by the
    correspondence:
    color <--> voltage
    The requirement that no adjacent colors are equal then corresponds to
    the requirement that no currents be zero in a solution to the circuit,
    while all voltages take on only 4 distinct values.
    You might look at circuits in which the voltages and currents are
    elements of fields other than the real numbers or the complex numbers.
    There is probably some interesting stuff out there, but after looking at
    it for an hour or so, I gave up for the time being.

    By the way, that N-dimensional fluid simulator I mentioned last time is
    here:

    http://www.xs4all.nl/~westy31/CellFlow/CellFlow.html

    Gerard
     
    Gerard Westendorp, Jun 7, 2008
    #4
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