Solving the Fokker-Planck Equation

VlaPy implements the Lenard-Bernstein [LB58] (LB) approximation and the Dougherty [Dou64] (DG) approximation of the Fokker-Planck equation. The LB approximation is given by

\[\frac{\partial f}{\partial t} = \nu_{ee} \frac{\partial}{\partial v} \left(v f + v_0^2 \frac{\partial f}{\partial v} \right)\]

where \(\nu_{ee}\) and \(v_0\) are the electron-electron collision frequency, and thermal velocity, respectively.

The DG approximation is given by

\[\frac{\partial f}{\partial t} = \nu_{ee} \frac{\partial}{\partial v} \left((v - \underline{v}) f + v_t^2 \frac{\partial f}{\partial v} \right)\]

where \(\underline{v} = \int f v dv\) and \(v_t^2 = \int f (v - \underline{v})^2 dv\) are the mean electron velocity, and the shifted thermal electron velocity.

Differencing Scheme

This operator is differenced backwards in time, and center differenced in velocity space, which gives

\[\frac{f^{n+1}_{\alpha} - f^{n}_{\alpha}}{\Delta t} = \nu_{ee} \left[f^{n+1}_\alpha + \bar{v}_\alpha \Delta_v(f^{n+1}_{\alpha}) + v_{rms}^2 \Delta^2_v(f^{n+1}_{\alpha})\right]\]

where \(\bar{v} = v, v_{rms}^2 = \int f v^2 dv\) for the LB operator and \(\bar{v} = v - \underline{v}, v_{rms}^2 = \int f \bar{v}^2 dv\) for the DG operator.

\[\Delta_v(f^{n+1}_{\alpha})= \frac{f^{n+1}_{\alpha+1} - f^{n+1}_{\alpha-1}}{2\Delta v}\]

and

\[\begin{split}\Delta^2_v(f^{n+1}_{\alpha})= \frac{-f^{n+1}_{\alpha+1} + 2f^{n+1}_{\alpha} - f^{n+1}_{\alpha-1}}{\Delta v^2} \\\end{split}\]

This system can be transformed into a linear system of equations in the form of

\[C_{ee} f^{n+1} = f^{n}\]

where \(C_{ee}\) is a matrix that corresponds to the finite difference operator stencil. In 1D, \(C_{ee}\) tridiagonal matrix. that can be solved directly.

Tests

This solver is tested to 1) return df/dt = 0 if a Maxwell-Boltzmann distribution is provided as input 2) conserve density, energy, and depending on the operator, velocity. 3) relax to an operator-dependent Maxwellian of the right temperature and drift velocity.

These tests are illustrated in notebooks/test_fokker_planck.ipynb and below:

Dou64

J. P. Dougherty. Model Fokker-Planck Equation for a Plasma and Its Solution. Physics of Fluids, 7(11):1788, 1964. URL: https://aip.scitation.org/doi/10.1063/1.2746779, doi:10.1063/1.2746779.

LB58

A. Lenard and Ira B. Bernstein. Plasma oscillations with diffusion in velocity space. Physical Review, 112(5):1456–1459, 1958. doi:10.1103/PhysRev.112.1456.