Report on the paper "ROM for non-linear conservative law" for group meeting
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\title[ROM]{Model Reduction for Nonlinear Conservative Law}
\author{T. Taddei, S. Perotto, A. Quarteroni}
\date{\today}
\begin{document}
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\begin{frame}{Background}
\begin{columns}
\column{0.5\textwidth} Let
$u(x, t) : [a, b] \times [0, T] \rightarrow \mathbb{R}$ solve
\[
\frac{du}{dt} + \frac{d}{dx} f(u) = 0
\]
where $f$ depends on a parameter $\mu$.
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\begin{itemize}
\item along the \emph{characteristic line} $x(t)$ defined by
\[
\frac{dx}{dt} = f'(u(x(0), 0))
\]
$u(x(t), t)$ is constant.
\item If two characteristic line crosses, a shock $x(t)$ is formed. It is determined by \emph{RH condition}:
\[
\frac{dx}{dt} = \frac{f(u_-) - f(u_+)}{u_- - u_+}
\]
\end{itemize}
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\begin{frame}{Assumptions}
\begin{itemize}
\item both sides $a \times [0, T]$, $b \times [0, T]$ are inflow boundaries.
\item only one shock is formed in the region $[a, b] \times [0, T]$.
\item continuous boundary conditions along $a \times [0, T]$, $b \times [0, T]$ and $[a, b] \times 0$.
\end{itemize}
Thus, a solution $u(x, t)$ can be be breaked down into three regular function, each defined on a region $T_i$ such that $\cup T_i = [a, b] \times [0, T]$.
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\begin{frame}{Partition}
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dotted line is the shock. Left: partition by the shock, right: partition by the characteristic line
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\begin{frame}[fragile]{Online Phase}
\begin{itemize}
\item For $\mu \in \{\mu_1, \mu_2, \ldots \mu_n\}$ do:
\item Lax-Friedrichs/Godunov monotone scheme to solve $u$
\item Detect shock by maximum change in discrete first derivative, $(x(t), t_0)$
\item Partition by the shock line to get three regular functions and store them in corresponding ROM model.
\end{itemize}
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\begin{frame}[fragile]{Offline Phase}
\begin{itemize}
\item For Query $\mu$ do:
\item polynomial interpolation to get $t_0$
\item Use Newton's method and characteristic line to find $u_-$ and $u_+$
\item Use Rankine-Hugoniot condition to get $x(t)$
\item Use ROM on each partitioned region to find interpolated functions.
\item Output the resulting function, combined from three functions.
\end{itemize}
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\begin{frame}[fragile]{Remark}
\begin{itemize}
\item Relies on assuming only a single shock line in $\mathbb{R}^2$. Generalize it to higher dimension is hard.
\item Different interpolation on region where function is regular.
\end{itemize}
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