Vlasov-Poisson system

The system considered is the two species Vlasov-Poisson system which is a simple model used to describe a collisionless plasma.

\[ \begin{cases} \partial_t f^+ + v \partial_xf^+ - \partial_x \phi \partial_v f^+ &= 0,\\ \partial_t f^- + v \partial_xf^- + \mu \partial_x \phi \partial_v f^- &= 0,\\ \end{cases}\]

where the potential $\phi := \phi(x)$ satisfy

\[ \partial_{xx} \phi = \int_{\mathbb{R}}(f^- - f^+) dv,\]

where $\mu > 0$ is the mass ratio, $f^+ := f^+(t,x,v)$ and $f^- := f^-(t,x,v)$ denote the distribution function for the ions and electrons respectively. The system is given with an initial condition

\[\begin{equation*} \begin{cases} f^+(0,x,v) = f_{in}^+(x,v),\\ f^-(0,x,v) = f_{in}^{-}(x,v),\\ \end{cases} \end{equation*}\]

on the domain $x \in [0, L]$, $v \in \mathbb{R}$ where we assume periodic boundary conditions in $x$ and vanishing boundary conditions in $v$.

The well-balanced numerical scheme is using a micro-macro type decomposition, where the macro part corresponds to the equilibrium and the micro part corresponds to the out of equilibrium part. The numerical solution will be written as $f^\pm = f_0 ^\pm + g^\pm$ where $f_0^\pm$ is a given equilibrium and $g^\pm$ can be seen as a perturbation of $f_0^\pm$ The unknown of the reformulated problem will be the perturbation $g^\pm$

To compute the initial solution, you can use the Coef type:

using Plots
using VlasovPoissonTwoSpecies

coef = Coef()

x_min, x_max, nx = 0., 1., 64
v_min, v_max, nv = -10., 10, 64

mesh_x = Mesh(x_min, x_max, nx)
mesh_v = Mesh(v_min, v_max, nv)

x = mesh_x.x
v = mesh_v.x

eq = EquilibriumManager(coef, mesh_x, mesh_v)

p = plot(layout=(2), xlabel="x", ylabel="v")
contourf!(p[1], x, v, eq.fe, title="fe")
contourf!(p[2], x, v, eq.fi, title="fi")