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

patankar eq

parent f01f2717
\newcommand {\ub} {\mathbf u}
\newcommand {\Fb} {\mathbf F}
\newcommand {\gb} {\tilde{\mathbf {g}}}
\newcommand {\nablab} {\bm{\nabla}}
\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
\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\\
%&= -\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\\
&= \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_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)
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