[repost] Re: Time Scale Calculus, Discrete Exterior Calculus, Navier-Stokes

rfamgm@xxxxxxxxx wrote:
[..]

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.

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.

[..]

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 ?

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.


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.

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

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