fokker_planck_test

Module for including functions used in testing the implementation of the the full-F Fokker-Planck collision operator.

Calculates the collisional flux $\Gamma_\|$ given Maxwellian input $F_s$ and $F_{s^\prime}$. The input Maxwellians are specified through their moments.

Calculates the collisional flux $\Gamma_\perp$ given Maxwellian input $F_s$ and $F_{s^\prime}$. The input Maxwellians are specified through their moments.

Calculates the fully expanded form of the collision operator $C_{s s^\prime}[F_s,F_{s^\prime}]$ given Maxwellian input $F_s$ and $F_{s^\prime}$. The input Maxwellians are specified through their moments.

Function calculating the fully expanded form of the collision operator taking as arguments the derivatives of $F_s$, $G_{s^\prime}$ and $H_{s^\prime}$. This function is designed to be used at the lowest level of a coordinate loop, with derivatives and integrals all previously calculated.

Function computing $ F_{\rm Maxwellian} $.

Function computing G, defined by

\[\nabla^4 G = -\frac{8}{\sqrt{\pi}} F \]

with

\[F = c_{\rm ref}^3 \pi^{3/2} F_{\rm Maxwellian} / n_{\rm ref} \]

the normalised Maxwellian. See Plasma Confinement, R. D. Hazeltine & J. D. Meiss, 2003, Dover Publications, pg 184, Chpt 5.2, Eqn (5.49).

Function computing H, defined by

\[\nabla^2 H = -\frac{4}{\sqrt{\pi}} F \]

with

\[F = c_{\rm ref}^3 \pi^{3/2} F_{\rm Maxwellian} / n_{\rm ref} \]

the normalised Maxwellian. See Plasma Confinement, R. D. Hazeltine & J. D. Meiss, 2003, Dover Publications, pg 184, Chpt 5.2, Eqn (5.49).

Calculates the collisional fluxes given input $F_s$ and $G_{s^\prime}$, $H_{s^\prime}$.

Function computing

\[\frac{\partial^2 F}{\partial v_\|^2}\]

for $ F = F_{\rm Maxwellian}$.

Function computing

\[\frac{\partial^2 F}{\partial v_\perp^2}.\]

for $ F = F_{\rm Maxwellian}$.

Function computing

\[\frac{\partial^2 F}{\partial v_\perp \partial v_\|}\]

for $ F = F_{\rm Maxwellian}$.

Function computing

\[\frac{\partial^2 G }{ \partial v_\|^2}\]

for Maxwellian input. See G_Maxwellian().

Function computing

\[\frac{\partial^2 G}{\partial v_\perp^2}\]

for Maxwellian input. See G_Maxwellian().

Function computing

\[\frac{\partial^2 G}{\partial v_\perp \partial v_\|}\]

for Maxwellian input. See G_Maxwellian().

Function computing

\[\frac{\partial F}{\partial v_\|}\]

for $ F = F_{\rm Maxwellian}$.

Function computing

\[\frac{\partial F}{\partial v_\perp}\]

for $ F = F_{\rm Maxwellian}$.

Function computing

\[\frac{\partial G}{\partial v_\perp}\]

for Maxwellian input. See G_Maxwellian().

Function computing

\[\frac{\partial H}{\partial v_\|}\]

for Maxwellian input. See H_Maxwellian().

Function computing

\[\frac{\partial H}{\partial v_\perp}\]

for Maxwellian input. See H_Maxwellian().

Function computing the normalised speed variable

\[\eta = \frac{\sqrt{(v_\| - u_\|)^2 + v_\perp^2}}{v_{\rm th}}\]

with $v_{\rm th} = \sqrt{2 p / n m}$ the thermal speed, and $p$ the pressure, $n$ the density and $m$ the mass.

Function to print the maximum error ${\rm MAX}(|f_{\rm numerical}-f_{\rm exact}|)$.

Function to print the maximum error ${\rm MAX}(|f_{\rm numerical}-f_{\rm exact}|)$ and the $L_2$ norm of the error

\[\sqrt{\int (f - f_{\rm exact})^2 v_\perp d v_\perp d v_\|/\int v_\perp d v_\perp d v_\|}.\]

Utility function that saves error data to a HDF5 file for later use.

Utility function that saves error data to a HDF5 file for later use.