From 01f47ace6756c28deb9ea0daaee3904ffa5ce9e0 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <jauslin@ias.edu>
Date: Thu, 11 Jan 2018 22:48:14 +0000
Subject: Initial commit

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+\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 {\it irreversible} Navier-Stokes equation in 2 dimensions
+\begin{equation}
+  \partial_tu=\nu\Delta u+g-\nabla w-(u\cdot\nabla)u,\quad
+  \nabla\cdot u=0
+  \label{ins}
+\end{equation}
+in which $g$ is the forcing term and $w$ is the pressure.
+We take periodic boundary conditions, so, at every given time, $u(t,\cdot)$ is a function on the unit torus $\mathbb T^2:=\mathbb R^2/\mathbb Z^2$. We represent $u(t,\cdot)$ using its Fourier series
+\begin{equation}
+  \hat u_k(t):=\int_{\mathbb T^2}dx\ e^{2i\pi kx}u(t,x)
+\end{equation}
+for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
+\begin{equation}
+  \partial_t\hat u_k=
+  -4\pi^2\nu k^2\hat u_k+\hat g_k-2i\pi k\hat w_k
+  -2i\pi\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{2i\pi k^\perp}{|k|}\hat\varphi_k\equiv\frac{2i\pi}{|k|}(-k_y\hat\varphi_k,k_x\hat\varphi_k)
+\end{equation}
+in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|}$,
+\begin{equation}
+  \partial_t\hat \varphi_k=
+  -4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
+  -\frac{2i\pi}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
+  (q\cdot p^\perp)(k^\perp\cdot q^\perp)\hat\varphi_p\hat\varphi_q
+  \label{ins_k}
+\end{equation}
+which, since $q\cdot p^\perp=q\cdot(k^\perp-q^\perp)=q\cdot k^\perp$, is
+\begin{equation}
+  \partial_t\hat \varphi_k=
+  -4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
+  +\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
+  (q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_p\hat\varphi_q}{|p||q|}
+  .
+  \label{ins_k}
+\end{equation}
+We truncate the Fourier modes and assume that $\hat\varphi_k=0$ 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 FFT}. We compute the last term in~\-(\ref{ins_k})
+\begin{equation}
+  T(\hat\varphi,k):=
+  \sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
+  \frac{\hat\varphi_p}{|p|}
+  (q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
+\end{equation}
+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\leqslant N_1,\ 0\leqslant n_2\leqslant 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\varphi,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)}
+  \frac{\hat\varphi_p}{|p|}
+  (q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
+\end{equation}
+provided
+\begin{equation}
+  N_i>4K_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 $p,q\in\mathcal K$, then $|p_i+q_i|\leqslant2K_i$. Therefore,
+\begin{equation}
+  T(\hat\varphi,k)
+  =
+  \frac1{N_1N_2}
+  \mathcal F^*\left(
+    \mathcal F(|p|^{-1}\hat\varphi_p)(n)
+    \mathcal F((|q|^{-1}(q\cdot k^\perp)(k\cdot q)\hat\varphi_q)_q)(n)
+  \right)(k)
+\end{equation}
+which we expand
+\begin{equation}
+  T(\hat\varphi,k)
+  =
+  \textstyle
+  \frac{k_x^2-k_y^2}{N_1N_2}
+  \mathcal F^*\left(
+    \mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
+    \mathcal F\left(\frac{q_xq_y}{|q|}\hat\varphi_q\right)(n)
+  \right)(k)
+  -
+  \frac{k_xk_y}{N_1N_2}
+  \mathcal F^*\left(
+    \mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
+    \mathcal F\left(\frac{q_x^2-q_y^2}{|q|}\hat\varphi_q\right)(n)
+  \right)(k)
+\end{equation}
+
+
+\vfill
+\eject
+
+\begin{thebibliography}{WWW99}
+\small
+\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{}
+\end{thebibliography}
+
+
+\end{document}
-- 
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