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Effects of Discretization
P249 - Fall 2010
10/12 - 10/14
Dan Fulton
Aliasing
Let force at x2 due to x1 be F(x1, x2).
Force invariant under
displacement.
x1
x2
F(x1, x2) = F(x2-x1)
Periodic variance depending on
grid location.
x1
x2
F(x1, x2) = F(x2-x1, .5(x2+x1))
Aliasing (cont.)
Aliasing (cont.)
In actuality, force at a given point is due to forces from all points in
the domain. Total force is the integral of forces acting from point to
point over the whole domain. We can also take the transform of the
total force.
So forces of wavenumber, k, are coupled (or aliased) to forces with
wavenumbers differing by an integer multiple of kg .
If |k| << |kg| then p=0 term will be largest.
Effects of Spatial Grid
We are dealing with charge, potential, field, and force in k-space.
The fourier transform of quantities on the grid are periodic as:
E(k-pkg) = E(k)
Intuitively, the severity of aliasing effect will depend on the shape
function, S(x) used to gather/scatter grid quantities.
(Birdsall 8.6 p164 for formal argument)
S(x)
xj-1
xj
xj+1
xj-1
xj
xj+1
xj-1
xj
xj+1
Effects of Finite Timesteps
QuickTime™ and a
decompressor
are needed to see this picture.
Dispersion Relation
For discrete time steps…
Dispersion Relation
Zero-order orbits.
Find fields along orbits and sub
back into difference eqns.
For magnetized plasma, including effects of both discrete
spatial and time steps.
Kinetic Theory of Fluctuation,
Noise, and Collisions
• Using PIC method, just have a sampling of particles.
• Understanding statistics of fluctuations in this sampling is
important.
Fluctuation Spectrum
Start with fourier transformed number density using a periodic
delta-function
To get fluctuating charge density and energy density spectrums:
Limiting Cases
• Fluctuation-dissipation theory (For Hamiltonian models)
• Spatial Spectrum (integrate prev. over
)
Limiting Cases (cont.)
•
•
•
Velocity Diffusion
With leapfrog integration we get…
Using the force along the unperturbed orbit for acceleration
we can calculate…
Velocity Drag
Find ensemble average change in velocity. Along zero-order orbit we
get 0. Evaluating next order we get:
Combining above with velocity diffusion, we can express as:
Distortion of plasma by test particle also creates drag. Treat plasma
as Vlasov gas and assume particle moves at constant speed.
Kinetic Equation
Combine velocity drag and diffusion terms in Fokker-Planck equation
to obtain total kinetic equation.