Homec4science
Diffusion COSOlver (new_coulomb) Browse

cosolver/doc/COSolver.org
b3f94e22e87cnew_coulomb

Last Change
Raw File

COSolver.org
View Options

#+TITLE: COllision SOlver - COSOlver
#+AUTHOR: Baptiste J. Frei
#+DATE: 20 Novembre 2019 (Creation date)
#+EMAIL: baptiste.frei@epfl.ch
#+LATEX_HEADER: \usepackage{amsmath}
* COSOlver - COllision SOlver
The COulomb SOlver is a FORTRAN program that computes the collisional
Hermite-Laguerre matrix of sophisticated linearized drift and gyrokientic
collision operator models. In particular, it contains the full linearized
Coulomb collision operator (valid for arbitrary mass and temperature ratios),
the Lorentz pitch-angle operator, and the Sugama
collision operator model. The Hermite-Laguerre moment expansion of these
collision operator models are based on a spherical tensorial harmonic
expansion of the Rosenbluth potentials. Then, a basis transformation is performed to obtain the moment
expansion of each collision operator models. In order to avoid numerical
error in computing large numerical values that arise in the moment
expansions, the arbitrary multiple precision FORTRAN library, FM library, is
used. The collisional matrices are output in H5 binary files in the iCa
dir. This document provides the essential definitions of the input parameter
and of the main COSOlver routines. Simple examples of collisional matrices
are shown.
* Collision Operator Models
We describe here the different collision operator models currently
implemented in the COSOlver code.
* Input Parameters
* Sampling Speed functions $\overline{\nu}_{ab*}^{Tpjgd}$
Here, we sample the test-particle speed function,
$\overline{\nu}_{ab*}^{Tpjgd}$ using the \verb|COSolver|. The test-particle
speed function is defined by
\begin{equation}
\overline{\nu}_{*ab}^{Tpjgd} = \sum_{j_1=0}^j L_{jj_1}^p \sum_{n=0}^2 \left[ a_{abn}^{pj_1} \alpha_{Eab}^{dj_1ngp} + a_{abn}^{'pj_1} \alpha_{eab}^{dj_1ngp}\right],
\end{equation}
\\
where we introduce the quantities
\begin{subequations}
\begin{equation}
\alpha_{Eab}^{djngp} &= \frac{4 \nu_{ab}}{\sqrt{\pi}} E_{ab}^{d+j-n +(g+p)/2},\\
\end{equation}
\begin{equation}
\alpha_{eab}^{djngp} &= \frac{4 \chi \nu_{ab}}{\sqrt{\pi}} e_{ab}^{d+j-n + (g+p)/2 +1},
\end{equation}
\end{subequations}
\\
and the mass and temperature dependent coefficients,
\begin{align}
a_{ab0}^{pl} &=-\frac{4 \sigma (\tau -1)}{\tau }\\
a^{pl}_{ab1} &= 4 \frac{\sigma}{\tau}l(\tau -2)-p (p -2 \sigma +4\frac{ \sigma}{\tau} +1)\\
a_{ab2}^{pl} & =\frac{\sigma}{\tau} \left(4 l^2+4 l (p-1)+\frac{3}{2} p(p-1) \right)
\end{align}
\\
and
\begin{align}
a^{'pl}_{ab0} &=\frac{4 (\tau -1) (\sigma +\tau )}{\tau },\\
a^{'pl}_{ab1} &=-2 (p+2 l) (\frac{\sigma}{\tau} (\tau -2)-1),\\
a_{ab2}^{'pl} & =- \frac{\sigma}{\tau} \left(4l^2+4 l (p-1)+\frac{3}{2}p (p-1) \right).
\end{align}
\\
We consider the like-species collisions, i.e. $a = b $, and set $\sigma = \tau
= \chi = 1$ and $\nu = 1$. We benchmark the \verb|COSolver| against Mathematica for the lowest
order test-particle speed functions (with correclty rounded numerical values).
| p | j | g | d | Mathematica | COSolver |
|---+---+---+---+--------------+------------|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | -1.59577 | -1.59577 |
| 0 | 1 | 0 | 0 | 0 | <E-55 |
| 2 | 2 | 2 | 0 | - 4.4881 | -4.488100 |
| 5 | 3 | 5 | 2 | 5260.06 | 5260.05656 |
We consider now electrion-ion collisions, i.e. $a =e$ and $b=i$, and set
$\sigma = m_e / m_i = 0.023^{2} $, $\tau = T_e/T_i =1 / 0.01$ (cold ion
limit). We benchmark the \verb|COSolver| against Mathematica for the lowest
order test-particle speed functions.
| p | j | g | d | Mathematica | COSolver |
|---+---+---+---+---------------+---------------------------|
| 0 | 0 | 0 | 0 | <E-19 | <E-59 |
| 1 | 0 | 1 | 0 | -2.25793 | -2.257934 |
| 0 | 1 | 0 | 0 | -0.00354 | <E-59 |
| 2 | 2 | 2 | 0 | -12.698664 | -12.698664 |
| 5 | 3 | 5 | 2 | 1523.81283446 | 1523.81283446119789691769 |
We note that it can be shown that $\overline{\nu}_{ab*}^{T0j00} =0$ for $j\leq
0$.
* Sampling Speed functions $\overline{\nu}_{ab*}^{Fpjgd}$
Here, we sample the back-reaction speed function,
$\overline{\nu}_{ab*}^{Fpjgd}$ using the \verb|COSolver|. The back-reaction speed function is defined by
\begin{equation}
\overline{\nu}_{*ab}^{Fpjgd} = \sum_{j_1=0}^{j} L_{jj_1}^p \sum_{n=0}^1 \left[ b_{abn}^{'pj_1} \beta_{eab}^{gpdnj_1} + b_{+abn}^{pj_1} \beta_{+eab}^{gpdnj_1} + b_{-abn}^{pj_1} \beta_{-Eeab}^{pdnj_1}\right],
\end{equation}
\\
where
\begin{align}
\beta_{eab}^{gpdnj_1} &= \frac{4 \nu_{ab}}{\sqrt{\pi}}\chi^{p+2j_1 +2n-1 }e_{ab}^{(g+p)/2 +1 + d+j_1+ n}, \\
\beta_{+eab}^{gpdnj_1} & = \frac{4 \nu_{ab}}{ \sqrt{\pi}}\chi^{p+2n -1}\sum_{j_2=0}^{j_1} \frac{\chi^{2j_2} j_1!}{j_2!} \frac{e_{ab}^{(p+g)/2 + n + 1 +d + j_2}}{2},\\
\beta_{-Eeab}^{pdnj_1} & = \frac{4 \nu_{ab}}{ \sqrt{\pi} }\chi^{2n - p-2}\left[ \frac{(p+j_1+1/2)!}{(1/2)!} \frac{E_{ab}^{d+n}}{2} \right. \nonumber \\
&\left. - \sum_{j_2=0}^{p+j_1} \frac{\chi^{2j_2 +1}(p+j_1+1/2)! }{(j_2 + 1/2)!} \frac{e_{ab}^{d+n+j_2+1}}{2}\right],
\end{align}
\\
with the mass and temperature dependent coefficients,
\begin{subequations} \label{eq:backreactioncoeff}
\begin{align}
b_{ab0}^{'pl} &= 4 \frac{\tau^2}{\sigma},\\
b_{ab1}^{'pl} &= - 8\frac{(p^2 + p -1)}{(2p-1)(2p+3)},\\
b_{+ab0}^{pl} & = -8 \frac{p(p-1)(l+1)}{(2p+1)(2p-1)} - \frac{\tau 8 ( 1 + p - \sigma p)}{\sigma (2p+1)},\\
b_{+ab1}^{pl} & =\frac{8(p+1)(p+2)}{(2p+1)(2p+3)},\\
b_{-ab0}^{pl} & = \frac{4(p+1)(p+2)(2p+2l+3)}{(2p+1)(2p+3)}+\frac{8 \chi^2(p - \sigma p - 1)}{(2p+1)} \\
b_{-ab1}^{pl} & = - 8 \frac{p(p-1)}{(2p+1)(2p-1)}
\end{align}
\end{subequations}
We consider the like-species collisions, i.e. $a = b $, and set $\sigma = \tau
= \chi = 1$ and $\nu = 1$. We benchmark the \verb|COSolver| against Mathematica for the lowest
order test-particle speed functions (with correclty rounded numerical values).
| p | j | g | d | Mathematica | COSolver |
|---+---+---+---+-------------+---------------------------------------|
| 0 | 0 | 0 | 0 | 0 | -.14960479781435801999493355277M-56 |
| 1 | 0 | 1 | 0 | 1.59577 | 1.59576912160573071176 |
| 0 | 1 | 0 | 0 | 0 | .13225011183238728846102158857948M-55 |
| 2 | 2 | 2 | 0 | 2.24405 | 2.24405032725805881341 |
| 5 | 3 | 5 | 2 | -158.354 | -158.35388220892584151737 |
We consider now electrion-ion collision, i.e. $a =e$ and $b=i$, and set
$\sigma = m_e / m_i = 0.023^{2} $, $\tau = T_e/T_i =1 / 0.01$ (cold ion
limit). We benchmark the \verb|COSolver| against Mathematica for the lowest
order test-particle speed functions.
| p | j | g | d | Mathematica | COSolver |
|---+---+---+---+---------------+----------------------------------------------|
| 0 | 0 | 0 | 0 | -4.51109 | -4.51109322249241617395 |
| 1 | 0 | 1 | 0 | 5.51899E-6 | .00000551899524534155 |
| 0 | 1 | 0 | 0 | -0.0000357952 | -.00003579533536315325 |
| 2 | 2 | 2 | 0 | -1.83481E-16 | -.00000000000000018349 |
| 5 | 3 | 5 | 2 | -4.1359E-24 | .107545698025060032726425929682195626215M-23 |
We note that it can be shown that $\overline{\nu}_{ab*}^{F0j00} =0$ for $j\leq
0$.
c4science · Help