KFE Absorbing Barrier Examples

[jlperla]

See #202

• [x] We should use the "extension" setup to manually generate the operators. Then those linear operators can be used in the DiffEq function. Worry about code clarity, not performance. (Resolved in 3d81888 by @chiyahn )
  • The first is: mu < 0, sigma = 0, basically just takes the interior of the extension (we thought)
  • Next, sigma > 0 with f(t,0) = 0 at the bottom, the reflecting barrier at the top
  • Next sigma > 0 with d_x f(t,0) = 0` at the bottom, the reflecting at the top.
• [x] Then solve all three of these in cells in the notebook for the PDE for t = [0, t_max] where t_max = -1/mu if mu < 0or someT` otherwise. (Resolved in 3d81888 by @chiyahn )
• [x] I will figure out the animation code given the solution to the PDE. (Resolved in e417521 by @chiyahn )
• [x] For all 3 of these, plot the S(t) calculation of the flux for the grid of t. See the formula given in the notebook, and keep in mind that this is a separate function/plot than the main animation. (Resolved in 69b45c1 by @chiyahn)
• [x] Fix the S(t) to use the 0 node and the forward different for the derivative. Using the boundary extrapolation, which will be BC specific. (Resolved by 76f0185)
• [x] Convert from using the bc version to manually fudging the extension ones and not using the extension. Come up with the boundary extrapolation function for each example. (Resolved in f90496e by @chiyahn )
• [x] Get the PDEs animating (Resolved in e417521 by @chiyahn )

Time dependent threshold

• [x] Examples with moving lower boundary (Resolved in decb3b6 by @chiyahn )

[jlperla]

Maybe install with

conda install -c conda-forge ffmpeg 

[jlperla]

@chiyahn

Here is the approach to animate: First, do that conda operation to install ffmpeg

Then,

using Plots
x = 1:5
show_every = 5
@gif for i=1:100
    plot(x, x .+ i/100 )
end every show_every

Basically, create that for loop to go through the solution to the differential equation calling plot, and the @gif will animate it.