A Discretization of the Navier-Stokes Equations

Note, 27 August 2009: This work has been reformatted for a narrower screen and minor typographical editing was done. The output was not regenerated.

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References

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Clebsch Coordinates

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Units and the Magnetic Induction

Please see my review of electro-magnetic units, which was done in order to obtain the equations for the magnetic induction vector. The review is here.

Two Clebsch Systems

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Tracing the Field Lines

The field lines were traced using a Taylor series. Please see section zero of the journal entry here.  Accuracy was achieved using a convergence process described in section 2 ('The Modules - get clebsch from spherical polar') of the journal entry here.  The last section of the just cited entry shows that a high degree of accuracy seems to have been achieved.  It seems to me that this is confirmed by the Christoffel symbol verification given below.

Navier-Stokes

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The Tensor Formalism

I don't define what a tensor is, I describe it ( here ).  I must say in connection with duality (e.g. see the quote from Levi-Civita, note[2] above, that follows immediately after Eq_1 here) that things (such as the Christoffel symbols and sets of partial derivatives) which are not tensors in the 'usual' sense of my description are handled in my formalism using the same Mathematica objects that are used to represent tensors.  There is, however, a more general tensor idea given by Sokolnikoff (note [5] above), and I recall applying this idea to the Christoffel symbols of the second kind with the conclusion that they are tensors in this more general sense.  I think it is worth while to quote Sokolnikoff at length:

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The Tensor Application

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The Second Kind Christoffel Symbols

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The Dual Second Kind Christoffel Symbols

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Set the Coordinate Variables

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The Differential Equation & Covariant Derivatives

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Numerical Differentiation

The Christoffel symbols and their derivatives are evaluated using a long process using their definition, but ultimately it is a matter of performing numerical differentiation of the coordinate transformations which were defined in the field line trace.  Formulae for this were derived using a multi-variate version of Taylor's theorem.  These formulae are also used to replace the derivatives of the unknown components of the velocity vector (vSUB[1] etc) with approximations involving the velocity at nearby points in the grid.

Check the Christoffel Symbol Calculation

The Christoffel symbol calculation can be checked by transforming to the spherical polar coordinate system, for which we have analytic expressions. The first cell shows the transformed values, and the second cell shows the analytical values. This seems to confirm both that the field line data and the numerical differentiation are very accurate.

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The Discretization

Mathematica's 4.1 and 6.0.2 took respectively about five and three minutes to do the discretization.  The output is non-linear because of the product terms in the differential equations. The output contains many unknowns because high order formulae were used in the numerical differentiation. In the output shown below, the derivatives have been left in their exact form. Output is only shown for the first of the three equations.

4.1

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6.0.2

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Converted by Mathematica      August 27, 2009