Commit c62e7f3b authored by Jonathan Lambrechts's avatar Jonathan Lambrechts
Browse files

patankar eq

parent f01f2717
\documentclass{paper}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{bm}
\usepackage[a4paper]{geometry}
\newcommand {\ub} {\mathbf u}
\newcommand {\Fb} {\mathbf F}
\newcommand {\gb} {\tilde{\mathbf {g}}}
\newcommand {\nablab} {\bm{\nabla}}
\begin{document}
\begin{align*}
\ub^{n+1}_p &= \ub_p^n+\frac{\Delta \Fb_p}{m}\\
\Fb_p &= -\gamma^n(\ub^{n+1}_p-\ub_f^{n+1}) + \gb m - V\nablab p
\end{align*}
\begin{align*}
\ub^{n+1}_p\left(\frac{m}{\Delta t}+\gamma^n\right)
&= \frac{m}{\Delta t}\ub^n_p+\gamma^n\ub^{n+1}_f+\gb m-V\nablab p\\
%\Fb_p
%&= -\gamma^n\left(-\ub_f^{n+1}+\frac{\Delta t}{m+\gamma^n\Delta t}\left(\frac{m}{\Delta t}\ub^n_p+\gamma^n\ub^{n+1}_f+\gb m-V\nablab p\right)\right)+\gb m -V\nablab p\\
\end{align*}
\begin{align*}
\Fb_p
&= \left(\frac 1{\gamma^n}+\frac{\Delta t}{m}\right)^{-1}\left(\ub_f^{n+1}-\ub_p^{n}+\frac{1}{\gamma^n}\left(\gb m-V\nablab p\right)\right)\\
\Fb_f
&= -\Fb_p +\gb m -V\nablab p\\
&= \left(\frac 1{\gamma^n}+\frac{\Delta t}{m}\right)^{-1}\left(\ub_p^n-\ub_f^{n+1}+ \frac{\Delta t}{m}\left(\gb m - V \nablab p \right)\right)
\end{align*}
\end{document}
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment