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SWIFT
SWIFTsim
Commits
09d36187
Commit
09d36187
authored
May 10, 2018
by
Josh Borrow
Browse files
Minor fixes to the formatting of equations in the theory for P-U
parent
c7960649
Changes
1
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Inline
Side-by-side
theory/SPH/Flavours/sph_flavours.tex
View file @
09d36187
...
...
@@ -386,14 +386,14 @@ evolution of the internal energy, as opposed to the entropy.
For P-U, the following choice of parameters in the formalism of
\S
\ref
{
sec:derivation
}
provides convenient properties:
\begin{align}
x
_
i =
&
(
\gamma
- 1) m
_
i u
_
i
~
,
\\
\tilde
{
x
}_
i =
&
1,
x
_
i =
&
~
(
\gamma
- 1) m
_
i u
_
i,
\\
\tilde
{
x
}_
i =
&
~
1,
\label
{
eq:sph:pu:xichoice
}
\end{align}
leading to the following requirements to ensure correct volume elements:
\begin{align}
y
_
i =
&
\sum
_{
j
}
(
\gamma
- 1) m
_
j u
_
j W
_{
ij
}
=
\bar
{
P
}_
i,
\\
\tilde
{
y
}_
i =
&
\sum
_{
j
}
W
_{
ij
}
=
\bar
{
n
}_
i,
\\
\tilde
{
y
}_
i =
&
\sum
_{
j
}
W
_{
ij
}
=
\bar
{
n
}_
i,
\label
{
eq:sph:pu:yichoice
}
\end{align}
with the resulting variables representing a smoothed pressure and particle
...
...
@@ -401,8 +401,8 @@ number density. This choice of variables leads to the following equation of
motion:
\begin{align}
\frac
{
\mathrm
{
d
}
\mathbf
{
v
}_
i
}{
\mathrm
{
d
}
t
}
= -
\sum
_
j (
\gamma
- 1)
^
2 m
_
j u
_
j u
_
i
&
\left
[
\frac
{
f
_{
ij
}}{
\bar
{
P
}_
i
}
\nabla
_
i W
_{
ij
}
(h
_
i) ~+
\right
.
\\
&
\frac
{
f
_{
ji
}}{
\bar
{
P
}_
j
}
\nabla
_
i W
_{
ji
}
(h
_
j) ~+
\\
&
\left
[
\frac
{
f
_{
ij
}}{
\bar
{
P
}_
i
}
\nabla
_
i W
_{
ij
}
(h
_
i) ~+
\right
.
\nonumber
\\
&
\frac
{
f
_{
ji
}}{
\bar
{
P
}_
j
}
\nabla
_
i W
_{
ji
}
(h
_
j) ~+
\nonumber
\\
&
\left
.
\nu
_{
ij
}
\bar
{
\nabla
_
i W
_{
ij
}}
\right
]~.
\label
{
eq:sph:pu:eom
}
\end{align}
...
...
@@ -411,9 +411,10 @@ the final term.
The
$
h
$
-terms are given as
\begin{align}
f
_{
ij
}
= 1 -
&
\left
[
\frac
{
h
_
i
}{
n
_
d (
\gamma
- 1)
\bar
{
n
}_
i
\left\{
m
_
j u
_
j
\right\}
}
\frac
{
\partial
\bar
{
P
}_
i
}{
\partial
h
_
i
}
\right
]
\times
\\
&
\left
( 1 +
\frac
{
h
_
i
}{
n
_
d
\bar
{
n
}_
i
}
\frac
{
\partial
\bar
{
n
}_
i
}{
\partial
h
_
i
}
\right
)
^{
-1
}
f
_{
ij
}
= 1 -
\left
[
\frac
{
h
_
i
}{
n
_
d (
\gamma
- 1)
\bar
{
n
}_
i
\left\{
m
_
j u
_
j
\right\}
}
\frac
{
\partial
\bar
{
P
}_
i
}{
\partial
h
_
i
}
\right
]
\left
( 1 +
\frac
{
h
_
i
}{
n
_
d
\bar
{
n
}_
i
}
\frac
{
\partial
\bar
{
n
}_
i
}{
\partial
h
_
i
}
\right
)
^{
-1
}
\label
{
eq:sph:pu:fij
}
\end{align}
with
$
n
_
d
$
the number of dimensions. In practice, the majority of
$
f
_{
ij
}$
is
...
...
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