Numerical scheme
This type contains data for the numerical scheme
\[f^{\pm}(t,x,v) = f_0^{\pm}(x,v) + g^{\pm}(t,x,v),\]
where $f_0^{\pm}(x,v)$ a stationary solution
\[T_{\phi} = v \partial_x - \partial_x \phi \partial_v\]
this term is computed in T_f
\[\begin{aligned} & \partial_t g^+ + T_{\phi_f} g^+ = \partial_x \phi_g \partial_v f_0^+,\\ & \partial_t g^{-} + \mu T_{- \phi_f} g^- = -\partial_x \phi_g \partial_v f_0^-, \end{aligned}\]
where $\phi_g := \phi_f- \phi_{f_0}$ is the potential associated with $g$. The two potentials satisfy the following equations
\[ \partial_{xx} \phi_f = \int_{\mathbb{R}}(f^- - f^+) dv, \quad \partial_{xx} \phi_g = \int_{\mathbb{R}}(g^- - g^+) dv,\]
and the initial condition for $g^{\pm}$ is given by
\[ \begin{aligned} g^+(0,x,v) = f_{in}^+(x,v) - f_0^+(x,v), \quad g^-(0,x,v) = f_{in}^-(x,v) - f_0^-(x,v). \end{aligned}\]
We use a time splitting scheme where the term $\pm\partial_x \phi_g \partial_v f_0^\pm$ is treated as a source term with three main steps (with $\mu=1$)
\[ \begin{aligned} &(1) \quad \partial_t g^{\pm} = \pm \partial_x \phi_g \partial_v f_0^\pm,\\ &(2) \quad \partial_t g^{\pm} + v \partial_x g^{\pm} = 0,\\ &(3) \quad \partial_t g^{\pm} \mp \partial_x \phi_f\partial_v g^{\pm} = 0. \end{aligned}\]
We consider a second order in time discretization to solve (2) and (3) from $t^n$ to $t^{n+1}$. In the following $\Delta t$ denotes the time step, $g^{n}$ the semi-discretization in time of the unknown $g$ at $t^n := n \Delta t$ and $g^{(k)}$ intermediate steps between $t^n$ and $t^{n+1}$. For simplicity we drop the index $\pm$ but it is important to note that the steps of the procedure described below are computed at the same time for $g^+$ and $g^-$. This is require to solve the Poisson equation when evaluating $\partial_x \phi_g$. One obtains the following algorithm
- Solve step (1) over a half time step with
compute_source: $g^{(1)} = g^n + \frac{\Delta t}{2} \partial_x \phi_{g^n} \partial_v f_0$ where $\partial_x\phi_{g^n}$ is calculated with Poisson equation, - Solve step (2) over a half time step: $g^{(2)}(x,v) = g^{(1)}(x-v\frac{\Delta t}{2}, v)$, with
advect. - Update the value of $E^{(2)} := -\partial_x \phi_{f^{(2)}}$ with Poisson solver and $f^{(2)} := f_0 + g^{(2)}$, with
compute_rhoandcompute_e. - Solve step (3) over a whole time step: $g^{(3)}(x,v) = g^{(2)}(x,v-E^{(2)} \Delta t)$
- Solve step (2) over a half time step: $g^{(4)}(x,v) = g^{3}(x-v\frac{\Delta t}{2},v)$
- Solve step (1) over a half time step: $g^{(5)} = g^{(4)} + \frac{\Delta t}{2} \partial_x \phi_{g^{(4)}} \partial_v f_0$ where $\partial_x\phi_{g^{(4)}}$ is calculated with Poisson solver.
# step 1
compute_source(scheme, 0.5dt)
# step 2
advect(scheme.advection_x, scheme.ge, v, 0.5dt)
advect(scheme.advection_x, scheme.gi, v, 0.5dt)
rho .= compute_rho(mesh_v, scheme.gi .- scheme.ge)
e .= compute_e(mesh_x, rho)
e .+= scheme.e_eq
# step 3
advect(scheme.advection_v, transpose(scheme.ge), -e, dt)
advect(scheme.advection_v, transpose(scheme.gi), e, dt)
# step 2
advect(scheme.advection_x, scheme.ge, v, 0.5dt)
advect(scheme.advection_x, scheme.gi, v, 0.5dt)
# step 1
compute_source(scheme, 0.5dt)
scheme.fe .= scheme.fe_eq .+ scheme.ge
scheme.fi .= scheme.fi_eq .+ scheme.gi