patankar eq

doc/Patankar.tex

+\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}