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Simpler expression for fft term

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Ian Jauslin
2018-01-12 19:20:59 +00:00
parent 7ee5507b93
commit cff1d2ee3c
3 changed files with 57 additions and 68 deletions
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45
docs/nstrophy_doc/nstrophy_doc.tex
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@ -49,16 +49,23 @@ 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}}
+\frac{4\pi^2}{|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
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 \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|}
\frac{(q\cdot p^\perp)|q|}{|p|}\hat\varphi_p\hat\varphi_q
.
\label{ins_k}
\end{equation}
@ -73,8 +80,7 @@ We truncate the Fourier modes and assume that $\hat\varphi_k=0$ if $|k_1|>K_1$ o
\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|}
\frac{(q\cdot p^\perp)|q|}{|p|}\hat\varphi_q\hat\varphi_p
\end{equation}
using a fast Fourier transform, defined as
\begin{equation}
@ -98,8 +104,7 @@ in which $\mathcal F^*$ is defined like $\mathcal F$ but with the opposite phase
\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|}
(q\cdot p^\perp)\frac{|q|}{|p|}\hat\varphi_q\hat\varphi_p
\end{equation}
provided
\begin{equation}
@ -109,29 +114,17 @@ Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\
\begin{equation}
T(\hat\varphi,k)
=
\textstyle
\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)
\mathcal F\left(\frac{p_x\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(q_y|q|\hat\varphi_q\right)(n)
-
\mathcal F\left(\frac{p_y\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(q_x|q|\hat\varphi_q\right)(n)
\right)(k)
\end{equation}
\bigskip
\vfill