From bca217e69837e2ecb788511b786f4adc9a74769e Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Tue, 11 Apr 2023 17:08:06 -0400 Subject: Remove extraneous directory in doc --- docs/nstrophy_doc/nstrophy_doc.tex | 336 ------------------------------------- 1 file changed, 336 deletions(-) delete mode 100644 docs/nstrophy_doc/nstrophy_doc.tex (limited to 'docs/nstrophy_doc/nstrophy_doc.tex') diff --git a/docs/nstrophy_doc/nstrophy_doc.tex b/docs/nstrophy_doc/nstrophy_doc.tex deleted file mode 100644 index 54e802e..0000000 --- a/docs/nstrophy_doc/nstrophy_doc.tex +++ /dev/null @@ -1,336 +0,0 @@ -\documentclass{ian} - -\usepackage{largearray} - -\begin{document} - -\hbox{} -\hfil{\bf\LARGE -{\tt nstrophy} -} -\vfill - -\tableofcontents - -\vfill -\eject - -\setcounter{page}1 -\pagestyle{plain} - -\section{Description of the computation} -\subsection{Irreversible equation} -\indent Consider the incompressible Navier-Stokes equation in 2 dimensions -\begin{equation} - \partial_tU=\nu\Delta U+G-(U\cdot\nabla)U,\quad - \nabla\cdot U=0 - \label{ins} -\end{equation} -in which $G$ is the forcing term. -We take periodic boundary conditions, so, at every given time, $U(t,\cdot)$ is a function on the torus $\mathbb T^2:=\mathbb R^2/(L\mathbb Z)^2$. We represent $U(t,\cdot)$ using its Fourier series -\begin{equation} - \hat U_k(t):=\frac1{L^2}\int_{\mathbb T^2}dx\ e^{i\frac{2\pi}L kx}U(t,x) -\end{equation} -for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as -\begin{equation} - \partial_t\hat U_k= - -\frac{4\pi^2}{L^2}\nu k^2\hat U_k+\hat G_k - -i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} - (q\cdot\hat U_p)\hat U_q - ,\quad - k\cdot\hat U_k=0 - \label{ins_k} -\end{equation} -We then reduce the equation to a scalar one, by writing -\begin{equation} - \hat U_k=\frac{i2\pi k^\perp}{L|k|}\hat u_k\equiv\frac{i2\pi}{L|k|}(-k_y\hat u_k,k_x\hat u_k) - \label{udef} -\end{equation} -in terms of which, multiplying both sides of the equation by $\frac L{i2\pi}\frac{k^\perp}{|k|}$, -\begin{equation} - \partial_t\hat u_k= - -\frac{4\pi^2}{L^2}\nu k^2\hat u_k - +\hat g_k - +\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} - \frac{(q\cdot p^\perp)(k^\perp\cdot q^\perp)}{|q||p|}\hat u_p\hat u_q - \label{ins_k} -\end{equation} -with -\begin{equation} - \hat g_k:=\frac{Lk^\perp}{2i\pi|k|}\cdot\hat G_k - . - \label{gdef} -\end{equation} -Furthermore -\begin{equation} - (q\cdot p^\perp)(k^\perp\cdot q^\perp) - = - (q\cdot p^\perp)(q^2+p\cdot q) -\end{equation} -and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore, -\begin{equation} - \partial_t\hat u_k= - -\frac{4\pi^2}{L^2}\nu k^2\hat u_k+\hat g_k - +\frac{4\pi^2}{L^2|k|}T(\hat u,k) - \label{ins_k} -\end{equation} -with -\begin{equation} - T(\hat u,k):= - \sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} - \frac{(q\cdot p^\perp)|q|}{|p|}\hat u_p\hat u_q - . - \label{T} -\end{equation} -We truncate the Fourier modes and assume that $\hat u_k=0$ if $|k_1|>K_1$ or $|k_2|>K_2$. Let -\begin{equation} - \mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\} - . -\end{equation} -\bigskip - -\point{\bf Reality}. -Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so -\begin{equation} - \hat u_{-k}=\hat u_k^* - . - \label{realu} -\end{equation} -Similarly, -\begin{equation} - \hat g_{-k}=\hat g_k^* - . - \label{realg} -\end{equation} -Thus, -\begin{equation} - T(\hat u,-k) - = - T(\hat u,k)^* - . - \label{realT} -\end{equation} -\bigskip - -\point{\bf FFT}. We compute T using a fast Fourier transform, defined as -\begin{equation} - \mathcal F(f)(n):=\sum_{m\in\mathcal N}e^{-\frac{2i\pi}{N_1}m_1n_1-\frac{2i\pi}{N_2}m_2n_2}f(m_1,m_2) -\end{equation} -where -\begin{equation} - \mathcal N:=\{(n_1,n_2),\ 0\leqslant n_1< N_1,\ 0\leqslant n_2< N_2\} -\end{equation} -for some fixed $N_1,N_2$. The transform is inverted by -\begin{equation} - \frac1{N_1N_2}\mathcal F^*(\mathcal F(f))(n)=f(n) -\end{equation} -in which $\mathcal F^*$ is defined like $\mathcal F$ but with the opposite phase. -\bigskip - -\indent The condition $p+q=k$ can be rewritten as -\begin{equation} - T(\hat u,k) - = - \sum_{p,q\in\mathcal K} - \frac1{N_1N_2} - \sum_{n\in\mathcal N}e^{-\frac{2i\pi}{N_1}n_1(p_1+q_1-k_1)-\frac{2i\pi}{N_2}n_2(p_2+q_2-k_2)} - (q\cdot p^\perp)\frac{|q|}{|p|}\hat u_q\hat u_p -\end{equation} -provided -\begin{equation} - N_i>3K_i. -\end{equation} -Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\%N_i$ (in which $\%N_i$ means `modulo $N_i$'), and, if $p,q,k\in\mathcal K$, then $|p_i+q_i-k_i|\leqslant3K_i$, so, as long as $N_i>3K_i$, then $(p_i+q_i-k_i)=0\%N_i$ implies $p_i+q_i=k_i$. -Therefore, -\begin{equation} - T(\hat u,k) - = - \textstyle - \frac1{N_1N_2} - \mathcal F^*\left( - \mathcal F\left(\frac{p_x\hat u_p}{|p|}\right)(n) - \mathcal F\left(q_y|q|\hat u_q\right)(n) - - - \mathcal F\left(\frac{p_y\hat u_p}{|p|}\right)(n) - \mathcal F\left(q_x|q|\hat u_q\right)(n) - \right)(k) -\end{equation} -\bigskip - -\point{\bf Energy}. -We define the energy as -\begin{equation} - E(t)=\frac12\int\frac{dx}{L^2}\ U^2(t,x)=\frac12\sum_{k\in\mathbb Z^2}|\hat U_k|^2 - . -\end{equation} -We have -\begin{equation} - \partial_t E=\int\frac{dx}{L^2}\ U\partial tU - = - \nu\int\frac{dx}{L^2}\ U\Delta U - +\int\frac{dx}{L^2}\ UG - -\int\frac{dx}{L^2}\ U(U\cdot\nabla)U - . -\end{equation} -Since we have periodic boundary conditions, -\begin{equation} - \int dx\ U\Delta U=-\int dx\ |\nabla U|^2 - . -\end{equation} -Furthermore, -\begin{equation} - I:=\int dx\ U(U\cdot\nabla)U - =\sum_{i,j=1,2}\int dx\ U_iU_j\partial_jU_i - = - -\sum_{i,j=1,2}\int dx\ (\partial_jU_i)U_jU_i - -\sum_{i,j=1,2}\int dx\ U_i(\partial_jU_j)U_i -\end{equation} -and since $\nabla\cdot U=0$, -\begin{equation} - I - = - -I -\end{equation} -and so $I=0$. -Thus, -\begin{equation} - \partial_t E= - \int\frac{dx}{L^2}\ \left(-\nu|\nabla U|^2+UG\right) - = - \sum_{k\in\mathbb Z^2}\left(-\frac{4\pi^2}{L^2}\nu k^2|\hat U_k|^2+\hat U_{-k}\hat G_k\right) - . -\end{equation} -Furthermore, -\begin{equation} - \sum_{k\in\mathbb Z^2}k^2|\hat U_k|^2\geqslant - \sum_{k\in\mathbb Z^2}|\hat U_k|^2-|\hat U_0|^2 - =2E-|\hat U_0|^2 -\end{equation} -so -\begin{equation} - \partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat U_0^2+\sum_{k\in\mathbb Z^2}\hat U_{-k}\hat G_k - \leqslant - -\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat U_0^2+ - \|\hat G\|_2\sqrt{2E} - . -\end{equation} -In particular, if $\hat U_0=0$ (which corresponds to keeping the center of mass fixed), -\begin{equation} - \partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E} - . -\end{equation} -Now, if $\frac{8\pi^2}{L^2}\nu\sqrt E<\sqrt2\|\hat G\|_2$, then -\begin{equation} - \frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}}\leqslant1 -\end{equation} -and so -\begin{equation} - \frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(t)})}{-\frac{4\pi^2}{L^2}\nu}\leqslant t+ - \frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(0)})}{-\frac{4\pi^2}{L^2}\nu} -\end{equation} -and -\begin{equation} - E(t) - \leqslant - \left( - \frac{L^2\sqrt2\|\hat G\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t}) - +e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)} - \right)^2 - . -\end{equation} -If $\frac{8\pi^2}{L^2}\nu\sqrt E>\sqrt2\|\hat G\|_2$, -\begin{equation} - \frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}}\geqslant1 -\end{equation} -and so -\begin{equation} - \frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(t)}-1)}{-\frac{4\pi^2}{L^2}\nu}\geqslant t+ - \frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(0)})-1}{-\frac{4\pi^2}{L^2}\nu} -\end{equation} -and -\begin{equation} - E(t) - \leqslant - \left( - \frac{L^2\sqrt2\|\hat G\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t}) - +e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)} - \right)^2 - . -\end{equation} -\bigskip - -\point{\bf Enstrophy}. -The enstrophy is defined as -\begin{equation} - \mathcal En(t)=\int\frac{dx}{L^2}\ |\nabla U|^2 - =\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat U_k|^2 - . -\end{equation} -\bigskip - -\point{\bf Numerical instability}. -In order to prevent the algorithm from blowing up, it is necessary to impose the reality of $u(x)$ by hand, otherwise, truncation errors build up, and lead to divergences. -It is sufficient to ensure that the convolution term $T(\hat u,k)$ satisfies $T(\hat u,-k)=T(\hat u,k)^*$. -After imposing this condition, the algorithm no longer blows up, but it is still unstable (for instance, increasing $K_1$ or $K_2$ leads to very different results). - -\subsection{Reversible equation} -\indent The reversible equation is similar to\-~(\ref{ins}) but instead of fixing the viscosity, we fix the enstrophy\-~\cite{Ga22}. -It is defined directly in Fourier space: -\begin{equation} - \partial_t\hat U_k= - -\frac{4\pi^2}{L^2}\alpha(\hat U) k^2\hat U_k+\hat G_k - -i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}} - (q\cdot\hat U_p)\hat U_q - ,\quad - k\cdot\hat U_k=0 -\end{equation} -where $\alpha$ is chosen such that the enstrophy is constant. -In terms of $\hat u$\-~(\ref{udef}), (\ref{gdef}), (\ref{T}): -\begin{equation} - \partial_t\hat u_k= - -\frac{4\pi^2}{L^2}\alpha(\hat u) k^2\hat u_k - +\hat g_k - +\frac{4\pi^2}{L^2|k|}T(\hat u,k) - . - \label{rns_k} -\end{equation} -To compute $\alpha$, we use the constancy of the enstrophy: -\begin{equation} - \sum_{k\in\mathbb Z^2}k^2\hat U_k\cdot\partial_t\hat U_k - =0 -\end{equation} -which, in terms of $\hat u$ is -\begin{equation} - \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\partial_t\hat u_k - =0 -\end{equation} -that is -\begin{equation} - \frac{4\pi^2}{L^2}\alpha(\hat u)\sum_{k\in\mathbb Z^2}k^4|\hat u_k|^2 - = - \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\hat g_k - +\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}|k|\hat u_k^*T(\hat u,k) -\end{equation} -and so -\begin{equation} - \alpha(\hat u) - =\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2} - . -\end{equation} -Note that, by\-~(\ref{realu})-(\ref{realT}), -\begin{equation} - \alpha(\hat u)\in\mathbb R - . -\end{equation} - - - -\vfill -\eject - -\begin{thebibliography}{WWW99} -\small -\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{} -\end{thebibliography} - -\end{document} -- cgit v1.2.3-70-g09d2