Homec4science
Diffusion COSOlver (improved_sugama) Browse

cosolver/doc/COSolver.tex~
3c0a5500b5baimproved_sugama

Last Change
Raw File

COSolver.tex~
View Options

% Created 2019-12-11 Wed 17:12
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{fixltx2e}
\usepackage{graphicx}
\usepackage{longtable}
\usepackage{float}
\usepackage{wrapfig}
\usepackage{soul}
\usepackage{textcomp}
\usepackage{marvosym}
\usepackage{wasysym}
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{hyperref}
\tolerance=1000
\usepackage{amsmath}
\providecommand{\alert}[1]{\textbf{#1}}
\title{COulomb SOlver}
\author{Baptiste J. Frei}
\date{20 Novembre 2019}
\hypersetup{
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs Org-mode version 7.9.3f}}
\begin{document}
\maketitle
\setcounter{tocdepth}{3}
\tableofcontents
\vspace*{1cm}
\section{Coulomb Solver}
\label{sec-1}
The COulomb SOlver is a programm that computes the collisional
Hermite-Laguerre matrix of the full linearized gyrokinetic and/or
drift-kinetic Coulomb Collision operator. This operator model is the full
linearized Landau collision operator valid for arbitrary temperature and mass
ratios. It is also accurate at arbitrary collisionality, conserves particles,
momentum, and energy. The Hermite-Laguerre moment expansion of this
linearized collision operator is based on a spherical tensorial harmonic
expansion of the Rosenbluth potential for both the test and field
(back-reaction) parts of the linearized Coulomb collision operator. The herein program provides also the collisional
Hermite-Laguerre matrix of the Lorentz pitch-angle scatering operator for
electrion-ion collision operator with the first order momentum conservative
term, and the ion-electorn collision operator in the small mass ration
expansion.
This document contains the routines definitions and analytical expressions
implemented in \verb|COSOlver|.
\begin{verbatim}
MODULE mymod
END MODULE mymod
\end{verbatim}
\section{Sampling Speed functions $\overline{\nu}_{ab*}^{Tpjgd}$}
\label{sec-2}
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).
\begin{center}
\begin{tabular}{rrrrrr}
p & j & g & d & Mathematica & COSolver \\
\hline
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & -1.59577 & -1.59577 \\
0 & 1 & 0 & 0 & 0 & <-55 \\
2 & 2 & 2 & 0 & - 4.4881 & -4.488100 \\
5 & 3 & 5 & 2 & 5260.06 & 5260.05656 \\
\end{tabular}
\end{center}
the 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.
\begin{center}
\begin{tabular}{rrrrrr}
p & j & g & d & Mathematica & COSolver \\
\hline
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 \\
\end{tabular}
\end{center}
We note that it can be shown that $\overline{\nu}_{ab*}^{T0j00} =0$ for $j\leq
0$.
\section{Sampling Speed functions $\overline{\nu}_{ab*}^{Fpjgd}$}
\label{sec-3}
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).
\begin{center}
\begin{tabular}{rrrrrr}
p & j & g & d & Mathematica & COSolver \\
\hline
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 \\
\end{tabular}
\end{center}
the 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.
\begin{center}
\begin{tabular}{rrrrrr}
p & j & g & d & Mathematica & COSolver \\
\hline
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 \\
\end{tabular}
\end{center}
We note that it can be shown that $\overline{\nu}_{ab*}^{F0j00} =0$ for $j\leq
0$.
\end{document}
c4science · Help