Vlasov Solver [work in progress]

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The first two lectures here http://wis.kuleuven.be/?lang=eng&page=colloq2007 look useful.
The first (Applications of hypercomplex analysis to the NavierStokes equations) might be a route to more stable simulations using quaternions.
The second (The challenge in the mathematics of space weather: Bridging the micromacro gap) offers a way of speeding up a simulation significantly.
The first (Applications of hypercomplex analysis to the NavierStokes equations) might be a route to more stable simulations using quaternions.
The second (The challenge in the mathematics of space weather: Bridging the micromacro gap) offers a way of speeding up a simulation significantly.
Ars artis est celare artem.

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More here http://www.pteponline.com/index_files/ ... 1207.PDF (An Exact Mapping from NavierStokes Equation to Schrödinger Equation via Riccati Equation)
Ars artis est celare artem.
Vlasov Solver [work in progress]
Has it been demonstrated that the 1/48 tesselation would contain all the electron orbits possible in a Vlasov metaequilibrium for the device? Does anyone have a rule for mapping the electron (or any other) gyroradius about each of the symmetry lines as a function of radial distance relative to the cube size?drmike wrote:One thing you might try is Indrek's symmetry method. Only 1/48 of the volume actually needs to be modeled if you assume symmetry. There are 3 cut planes which give you 1/8th of a volume, then 3 cut planes through the corners that give you 1/6th of that. If you take the derivatives to be zero at the interfaces, you can fold the whole system around to get a final picture.
That factor of 48 can help your phase space tremendously. Same storage, more accuracy.
Bob Terry

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 Posts: 815
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Quaternion calculations are generally better behaved mathematically, which is why they're now used by NASA and in 3D computer games. They put less stress on your resources and give you better performance.reterry wrote:So how would quaternions help here?alexjrgreen wrote:Did I mention quaternions...?KitemanSA wrote:Some have postulated in these pages that rotation is a significant effect, which this process assumes away.
More interestingly, expressing the underlying physics in quaternions can result in fewer assumptions about the kind of solutions you expect.
Physics calculations are almost always constrained by the resources available, and so are critically dependant on simplifying assumptions. Whether these assumptions are valid is a substantial discussion, one that is not always revisited when tackling a new problem with a standard formula.
A long time ago, I solved Rubik's Cube on a Sinclair ZX80 with 1Kb of memory. For the curious, I treated the edges and corners completely separately, ignored the orientation and looked for sequences that returned the pieces to their original position. It took two whole weeks to run and I didn't find all the moves (some were excluded by my assumptions), but I found enough to solve the cube inefficiently from any position.
The Maxwell equations that are taught in schools are different from the ones that Maxwell published. To make it easier to find solutions, Heavyside simplified them and we've been using his version ever since. As a result, there are behaviours in electromagnetic systems that we neglect to look for.
Maxwell's original theory used quaternions. One of the papers I linked to on another thread http://www.pteponline.com/index_files/ ... 1703.PDF uses them to discuss the possibility of scalar waves. Heavyside's version ignores such solutions, and I don't think anyone knows whether they really exist or not. If they did, though, you might expect them to show up as instabilities in plasma...
Ars artis est celare artem.
Thanks for the pointer to the quaternian paper.
There's a typo in equation 14. The + in front of the v/c^2 should be a  (and that is consistent with the rest of that section).
The end of section 3 is kind of dubious to me. The authors use the same vector potential to represent two different things, and then sum them. This needs a bit more elaboration, it may well be correct, but it is not well spelled out.
Then in section 4 they make a huge leap that I really have to take issue with. Equations 25, 26 and 27 should really be two equations which are derived from 24. The real portion of 24 is one equation, and the imaginary portion is the second. There are not 3 separate equations.
The authors assumed that the original form of the continuity equation was valid, and simply proceeded from there. I don't think that makes sense  if they want to prove that quaternian form delivers new observable reality, they need to follow the math correctly.
What makes sense from what they have is that div(J) + d(rho)/dt = curl(J) and grad(rho) + 1/c^2 d(J)/dt = 0. The wave equations they get don't make sense  charge is not a field moving at light speed. It would be interesting to see how the actual quaternian equations turn into something resembling what we actually observe.
(Edited since I got a chance to read part of the article)
There's a typo in equation 14. The + in front of the v/c^2 should be a  (and that is consistent with the rest of that section).
The end of section 3 is kind of dubious to me. The authors use the same vector potential to represent two different things, and then sum them. This needs a bit more elaboration, it may well be correct, but it is not well spelled out.
Then in section 4 they make a huge leap that I really have to take issue with. Equations 25, 26 and 27 should really be two equations which are derived from 24. The real portion of 24 is one equation, and the imaginary portion is the second. There are not 3 separate equations.
The authors assumed that the original form of the continuity equation was valid, and simply proceeded from there. I don't think that makes sense  if they want to prove that quaternian form delivers new observable reality, they need to follow the math correctly.
What makes sense from what they have is that div(J) + d(rho)/dt = curl(J) and grad(rho) + 1/c^2 d(J)/dt = 0. The wave equations they get don't make sense  charge is not a field moving at light speed. It would be interesting to see how the actual quaternian equations turn into something resembling what we actually observe.
(Edited since I got a chance to read part of the article)
Geometric Algebra
Quaternians are really a special case of Geometric Algebra, as are complex numbers.
Actually, I have come to the conclusion that just about everything I have ever done in Physics I represented and manipulated using an inferior special case of Geometric Algebra.
GA provides concepts that work in any number of dimensions, including a generalisation of curl that applies in any number of dimensions. Algebraic statements are always true in GA, regardless if you are working with an underlying hyperbolic, spherical, or whatever geometry, and the number of dimensions you are working in. Everything always has a geometric meaning that you can understand, regardless of the number of dimensions. Also, it is easy to do everything in a coordinate free manner.
Generally you are actually working in a higher dimension than the problem you are solving, and the extra dimensions transparently make it easier to express what you need. Quaternians hint at this.
Maxwell's equations are a single equation in GA: nabla F = J
It's a useful equation because the dellike operator is invertible in GA.
A good primer is for 3D graphics work is: http://www.jaapsuter.com/paper/ga_primer.pdf
An intro aimed more at physicists: http://www.mrao.cam.ac.uk/~clifford/int ... intro.html
People interested in Quantum Mechanics would probably be interested in: http://modelingnts.la.asu.edu/html/GAinQM.html
David Hestenes has done a lot of pioneering work on GA and expressing physics using GA. I suggest doing a search on him and GA. I wish I could find his books for prices I could afford though.
I am just starting to put together the sources to work this stuff out, so I am not much of a resource myself, I'm afraid. I do think GA worth looking into though.
Actually, I have come to the conclusion that just about everything I have ever done in Physics I represented and manipulated using an inferior special case of Geometric Algebra.
GA provides concepts that work in any number of dimensions, including a generalisation of curl that applies in any number of dimensions. Algebraic statements are always true in GA, regardless if you are working with an underlying hyperbolic, spherical, or whatever geometry, and the number of dimensions you are working in. Everything always has a geometric meaning that you can understand, regardless of the number of dimensions. Also, it is easy to do everything in a coordinate free manner.
Generally you are actually working in a higher dimension than the problem you are solving, and the extra dimensions transparently make it easier to express what you need. Quaternians hint at this.
Maxwell's equations are a single equation in GA: nabla F = J
It's a useful equation because the dellike operator is invertible in GA.
A good primer is for 3D graphics work is: http://www.jaapsuter.com/paper/ga_primer.pdf
An intro aimed more at physicists: http://www.mrao.cam.ac.uk/~clifford/int ... intro.html
People interested in Quantum Mechanics would probably be interested in: http://modelingnts.la.asu.edu/html/GAinQM.html
David Hestenes has done a lot of pioneering work on GA and expressing physics using GA. I suggest doing a search on him and GA. I wish I could find his books for prices I could afford though.
I am just starting to put together the sources to work this stuff out, so I am not much of a resource myself, I'm afraid. I do think GA worth looking into though.
http://www.jaapsuter.com/paper/ga_primer.pdf is a bad link. It returns a blank page.

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Re: Geometric Algebra
Some useful stuff here: http://www.mrao.cam.ac.uk/~clifford/ptI ... /course99/Ray wrote:Quaternians are really a special case of Geometric Algebra, as are complex numbers.
Ars artis est celare artem.
And if you really want to go nuts, check out the Journal of Geometry and Physics.
Clifford Algebras and Manifolds are also good things to look into. N dimensional spaces seem complicated, but they tie a lot of complexity together for better understanding overall.
Clifford Algebras and Manifolds are also good things to look into. N dimensional spaces seem complicated, but they tie a lot of complexity together for better understanding overall.

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Doug Sweetser reckons that quaternions are adequate.drmike wrote:And if you really want to go nuts, check out the Journal of Geometry and Physics.
Either way, the point is to drop Heavyside's oversimplification of Maxwell.
The solver should avoid assumptions that exclude possible real solutions.
Ars artis est celare artem.