From 2a24406ca6c64901100e982b45f8e7fb913864c3 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Thu, 26 May 2022 14:12:18 -0400
Subject: Add stability comment in doc

---
 docs/nstrophy_doc/nstrophy_doc.tex | 6 ++++++
 1 file changed, 6 insertions(+)

(limited to 'docs/nstrophy_doc/nstrophy_doc.tex')

diff --git a/docs/nstrophy_doc/nstrophy_doc.tex b/docs/nstrophy_doc/nstrophy_doc.tex
index fd609a9..418520f 100644
--- a/docs/nstrophy_doc/nstrophy_doc.tex
+++ b/docs/nstrophy_doc/nstrophy_doc.tex
@@ -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
   .
 \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
 \eject
-- 
cgit v1.2.3-70-g09d2

