Commit c62e7f3b by Jonathan Lambrechts

### patankar eq

parent f01f2717
File moved
doc/Patankar.tex 0 → 100644
 \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}
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