software/Nstrophy
1
Fork 0
Code Issues Pull Requests Releases Activity

Proper normalization for enstrophy norm

Browse Source
This commit is contained in:
Ian Jauslin
2023-06-13 19:01:37 -04:00
parent 34b7a0c277
commit 3b58896e3b
2 changed files with 18 additions and 5 deletions
Download Patch File Download Diff File Expand all files Collapse all files

13
docs/nstrophy_doc.tex
View File

@ -174,12 +174,23 @@ The choice of the norm $\|\cdot\|$ matters.
\begin{equation}
\|f\|:=\frac1{\mathcal N}\sqrt{\sum_k k^2|f_k|^2}
\quad
\mathcal N:=\sqrt{\sum_k k^2|\hat u_k^{(n)}|^2}
\mathcal N:=\left(\frac{\sqrt{\sum_k k^2|\hat u_k^{(n)}|^2}+\sqrt{\sum_k k^2|\hat U_k^{(n)}|^2}}{\sum_k k^2|\hat u_k^{(n)}|^2}\right)^{\frac13}
.
\end{equation}
Doing so controls the error of the enstrophy through
\begin{equation}
\mathcal N^2|\mathcal En(\hat u)-\mathcal En(\hat U)|\equiv|\|\hat u\|^2-\|\hat U\|^2|\leqslant \|\hat u-\hat U\|(\|\hat u\|+\|\hat U\|)
\end{equation}
so
\begin{equation}
\mathcal N^2
|\mathcal En(\hat u)-\mathcal En(\hat U)|\leqslant
\|\hat u-\hat U\|\frac1{\mathcal N}\left(\sqrt{\sum_k k^2|\hat u_k|^2}+\sqrt{\sum_k k^2|\hat U_k|^2}\right)
\end{equation}
and thus
\begin{equation}
\frac{|\mathcal En(\hat u)-\mathcal En(\hat U)|}{\mathcal En(\hat u)}\leqslant
\|\hat u-\hat U\|
.
\end{equation}
\end{itemize}