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For time dependent 3-dim vectors [math] {\bf v} , {\bf b} [/math] and constant c,
I'm dealing with an differential equation of the form
[math] \dfrac{d}{dt} {\bf v} = c {\bf b} + {\bf b} \times {\bf v} [/math]
Does anyone recognize this type form any other setting?
For c=0 we have that [math] L = {\bf b} \times [/math] is just a linear map and I know that
[math] \dfrac{d}{dt} {\bf v} = L {\bf v} [/math]
is solved via
https://en.wikipedia.org/wiki/Magnus_expansion
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Mostly, I want to know what known discretization schemes available. For the first equation. Maybe there's something in fluid-dynamics, hydro-dynamics or electromagnetic field theory like that. (E.g. a text on the discretization of the Newton equation for the Lorentz force)
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>>8607966
How about writing it using 4x4 matrix L. Append a 1 to v to make it 4x1. Use Magnus again
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>>8608350
This.
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Kind of looks like the Landau-Lifshitz-Gilbert equation for magnetodynamics if you define v as M and b as (M x H_eff). There are some notes on it on the internet for numerical solutions. Hope this helps!
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>>8608384
wow dude what is your education level if you know this kind of stuff ?
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>>8608387
My professor mentioned it during undergrad E&M. I think it's important for spintronics or something, but I used it to show that it can explain how an MRI works in a presentation for that class.
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>>8608350
>Append a 1 to v to make it 4x1.
Maybe I'm confused, but since this an ODE and v is not a constant, I'd have [math] \frac{d}{dt}v_4 = v_4 [/math] and [math] v_4 = 1 [/math] would be merely an initial condition.
Or do you mean to restrict the differential operator somehow? But what's the integration procedure, then?
>>8608384
Thanks, I'll try to find a text on it. Although b being a function of v (what we have with this magneto-equation) is a more complicated case.
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Also, does anybody know how well integration methods such as Runge Kutta to various degrees carry over to 3-dimensional system?
Does anybody have a reference for e.g. Runge-Kutta in those dimensions?
https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods
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>>8608858
Make L 4x4 with the bottom row all zeros and the first 3 rows of the last column equal to b and the top left 3x3 equal to the matrix form of b x.
Take v and make it 4x1 by adding a 1 to the end. Then d/dt v = L v does the job.
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>>8609832
I think that makes sense to me, thx, will look at it.
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