Time Scale Calculus, Discrete Exterior Calculus, Navier-Stokes
rfa...@gmail.com
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
Gerard Westendorp
[..]
> 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
caspro
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 ?
Gerard Westendorp
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