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Select norm in parameter

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Ian Jauslin
2023-06-13 23:56:35 -04:00
parent 3b58896e3b
commit 251426aaaa
6 changed files with 112 additions and 24 deletions
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docs/nstrophy_doc.tex
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@ -145,6 +145,7 @@ To be safe, we also set a maximal value for $\delta$ via the {\tt max\_delta} pa
\indent
The choice of the norm $\|\cdot\|$ matters.
It can be made by specifying the parameter {\tt adaptive\_norm}.
\begin{itemize}
\item A naive choice is to take $\|\cdot\|$ to be the normalized $L_1$ norm:
\begin{equation}
@ -154,6 +155,7 @@ The choice of the norm $\|\cdot\|$ matters.
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|
.
\end{equation}
This norm is selected by choosing {\tt adaptive\_norm=L1}.
\item Empirically, we have found that $|\hat u-\hat U|$ behaves like $k^{-3}$ for {\tt RKDP54} and {\tt RKF45}, and like $k^{-\frac32}$ for {\tt RKBS32}, so a norm of the form
\begin{equation}
@ -168,6 +170,7 @@ The choice of the norm $\|\cdot\|$ matters.
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|k^{-\frac32}
\end{equation}
are sensible choices.
These norms are selected by choosing {\tt adaptive\_norm=k3} and {\tt adaptive\_norm=k32} respectively.
\item
Another option is to define a norm based on the expression of the enstrophy\-~(\ref{enstrophy}):
@ -193,6 +196,7 @@ The choice of the norm $\|\cdot\|$ matters.
\|\hat u-\hat U\|
.
\end{equation}
This norm is selected by choosing {\tt adaptive\_norm=enstrophy}.
\end{itemize}
\bigskip