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Add stability comment in doc

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Ian Jauslin 2022-05-26 14:12:18 -04:00
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docs/nstrophy_doc/nstrophy_doc.tex
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@ -236,6 +236,12 @@ The enstrophy is defined as
=\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat u_k|^2 =\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat u_k|^2
. .
\end{equation} \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 $T(\hat\varphi,k)$ satifies $T(\hat\varphi,-k)=T(\hat\varphi,k)^*$.
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).
\vfill \vfill
\eject \eject