From 2a24406ca6c64901100e982b45f8e7fb913864c3 Mon Sep 17 00:00:00 2001 From: Ian Jauslin 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(+) 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 v1.2.3-54-g00ecf