timedomainresonace/model.lyx

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\begin_body

\begin_layout Title
1D Euler time domain
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\frac{\partial\rho}{\partial t}+\frac{\partial m}{\partial x} & = & 0\\
\frac{\partial m}{\partial t}+\frac{\partial\left(\rho u^{2}+p_{0}+p\right)}{\partial x} & = & 0\\
\frac{\partial E}{\partial t}+\frac{\partial\left[\left(E+p\right)u\right]}{\partial x} & = & 0
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
with
\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset Formula $\rho e=\rho c_{v}T=\frac{c_{v}}{R}\rho R_{s}T=\frac{c_{v}}{R}\left(p_{0}+p\right)=\frac{1}{\gamma-1}\left(p_{0}+p\right)$
\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
E=\rho e+\rho\tfrac{1}{2}u^{2}=\frac{p_{0}+p}{\left(\gamma-1\right)}+\rho\tfrac{1}{2}u^{2}
\]

\end_inset


\end_layout

\begin_layout Standard
If we replace 
\begin_inset Formula $E$
\end_inset

 with
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
E=\hat{E}+\frac{p_{0}}{\gamma-1}
\end{equation}

\end_inset

then
\begin_inset Formula 
\[
\frac{\partial\hat{E}}{\partial t}+\frac{\partial\left(\hat{E}+p_{0}+p\right)u}{\partial x}=0
\]

\end_inset

so 
\begin_inset Formula $E$
\end_inset

 is internal energy per unit volume!
\end_layout

\begin_layout Section
Scheme
\end_layout

\begin_layout Standard
If we say:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\frac{\partial U}{\partial t}+\frac{\partial}{\partial x}\left(F(U)\right)=0
\]

\end_inset

which is
\begin_inset Formula 
\begin{equation}
\frac{U_{i}^{n+1}-\left(\frac{1}{2}\left(U_{i+1}+U_{i-1}^{^{2}}\right)\right)}{\Delta t}+\frac{F(U_{i+1}^{n})-F(U_{i-1}^{n})}{2\Delta x}=0
\end{equation}

\end_inset

in its discrete form
\end_layout

\begin_layout Standard
then
\begin_inset Formula 
\[
U=\left\{ \begin{array}{c}
\rho\\
\rho u\\
\rho E
\end{array}\right\} 
\]

\end_inset

and Lax-Friedrichs says for a middle node
\begin_inset Formula 
\[
U_{i}^{n+1}=\frac{1}{2}\left(U_{i+1}^{n}+U_{i-1}^{n}\right)-\lambda\left(F\left(U_{i+1}^{n}\right)-F(U_{i-1}^{n})\right)
\]

\end_inset

with
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\lambda=\frac{\Delta t}{2\Delta x}
\]

\end_inset

Total domain length: 
\begin_inset Formula $L$
\end_inset

.
 If number of gridpoints is 3, one left, one right, than 
\begin_inset Formula $dx=L/(gp-1$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
L
\]

\end_inset


\end_layout

\begin_layout Section
Right wall bc
\end_layout

\begin_layout Standard
At a wall:
\begin_inset Formula 
\[
m=0
\]

\end_inset

and
\begin_inset Formula 
\[
\frac{\partial}{\partial x}\left(F(U)\right)\approx\frac{F\left(U_{i}^{n}\right)-F(U_{i-1}^{n})}{\Delta x}
\]

\end_inset

but
\end_layout

\begin_layout Section
Left pressure bc
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\frac{\partial}{\partial x}\left(F(U)\right)\approx\frac{F\left(U_{i+1}^{n}\right)-F(U_{i}^{n})}{\Delta x}
\]

\end_inset

for continuity, energy for momentum:
\begin_inset Formula 
\[
\frac{\partial}{\partial x}\left(F(U)\right)\approx\frac{\left[\rho u^{2}+p\right]_{i+1}-(\rho u^{2})|_{i}-P_{pres}}{\Delta x}
\]

\end_inset


\end_layout

\end_body
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