147 lines
4.4 KiB
TeX
147 lines
4.4 KiB
TeX
\documentclass{ian}
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\usepackage{largearray}
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\begin{document}
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\hbox{}
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\hfil{\bf\LARGE
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{\tt nstrophy}
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}
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\vfill
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\tableofcontents
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\vfill
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\eject
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\setcounter{page}1
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\pagestyle{plain}
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\section{Description of the computation}
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\subsection{Irreversible equation}
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\indent Consider the {\it irreversible} Navier-Stokes equation in 2 dimensions
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\begin{equation}
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\partial_tu=\nu\Delta u+g-\nabla w-(u\cdot\nabla)u,\quad
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\nabla\cdot u=0
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\label{ins}
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\end{equation}
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in which $g$ is the forcing term and $w$ is the pressure.
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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
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\begin{equation}
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\hat u_k(t):=\int_{\mathbb T^2}dx\ e^{2i\pi kx}u(t,x)
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\end{equation}
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for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
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\begin{equation}
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\partial_t\hat u_k=
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-4\pi^2\nu k^2\hat u_k+\hat g_k-2i\pi k\hat w_k
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-2i\pi\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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(q\cdot\hat u_p)\hat u_q
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,\quad
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k\cdot\hat u_k=0
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\label{ins_k}
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\end{equation}
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We then reduce the equation to a scalar one, by writing
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\begin{equation}
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\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)
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\end{equation}
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in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|}$,
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\begin{equation}
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\partial_t\hat \varphi_k=
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-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
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-\frac{2i\pi}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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(q\cdot p^\perp)(k^\perp\cdot q^\perp)\hat\varphi_p\hat\varphi_q
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\label{ins_k}
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\end{equation}
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which, since $q\cdot p^\perp=q\cdot(k^\perp-q^\perp)=q\cdot k^\perp$, is
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\begin{equation}
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\partial_t\hat \varphi_k=
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-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
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+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_p\hat\varphi_q}{|p||q|}
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.
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\label{ins_k}
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\end{equation}
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We truncate the Fourier modes and assume that $\hat\varphi_k=0$ if $|k_1|>K_1$ or $|k_2|>K_2$. Let
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\begin{equation}
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\mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\}
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.
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\end{equation}
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\bigskip
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\point{\bf FFT}. We compute the last term in~\-(\ref{ins_k})
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\begin{equation}
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T(\hat\varphi,k):=
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\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
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\frac{\hat\varphi_p}{|p|}
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(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
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\end{equation}
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using a fast Fourier transform, defined as
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\begin{equation}
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\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)
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\end{equation}
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where
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\begin{equation}
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\mathcal N:=\{(n_1,n_2),\ 0\leqslant n_1\leqslant N_1,\ 0\leqslant n_2\leqslant N_2\}
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\end{equation}
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for some fixed $N_1,N_2$. The transform is inverted by
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\begin{equation}
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\frac1{N_1N_2}\mathcal F^*(\mathcal F(f))(n)=f(n)
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\end{equation}
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in which $\mathcal F^*$ is defined like $\mathcal F$ but with the opposite phase.
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\bigskip
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\indent The condition $p+q=k$ can be rewritten as
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\begin{equation}
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T(\hat\varphi,k)
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=
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\sum_{p,q\in\mathcal K}
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\frac1{N_1N_2}
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\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)}
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\frac{\hat\varphi_p}{|p|}
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(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
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\end{equation}
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provided
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\begin{equation}
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N_i>4K_i.
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\end{equation}
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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,
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\begin{equation}
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T(\hat\varphi,k)
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=
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\frac1{N_1N_2}
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\mathcal F^*\left(
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\mathcal F(|p|^{-1}\hat\varphi_p)(n)
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\mathcal F((|q|^{-1}(q\cdot k^\perp)(k\cdot q)\hat\varphi_q)_q)(n)
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\right)(k)
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\end{equation}
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which we expand
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\begin{equation}
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T(\hat\varphi,k)
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=
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\textstyle
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\frac{k_x^2-k_y^2}{N_1N_2}
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\mathcal F^*\left(
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\mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
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\mathcal F\left(\frac{q_xq_y}{|q|}\hat\varphi_q\right)(n)
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\right)(k)
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-
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\frac{k_xk_y}{N_1N_2}
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\mathcal F^*\left(
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\mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
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\mathcal F\left(\frac{q_x^2-q_y^2}{|q|}\hat\varphi_q\right)(n)
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\right)(k)
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\end{equation}
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\vfill
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\eject
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\begin{thebibliography}{WWW99}
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\small
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\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{}
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\end{thebibliography}
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\end{document}
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