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SWIFT
SWIFTsim
Commits
c7960649
Commit
c7960649
authored
May 10, 2018
by
Josh Borrow
Browse files
Added basic theory (equations essentially) behind PUSPH here
parent
5779ee54
Changes
2
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theory/SPH/Flavours/sph_flavours.tex
View file @
c7960649
...
@@ -377,7 +377,70 @@ itself is \emph{not} updated.
...
@@ -377,7 +377,70 @@ itself is \emph{not} updated.
\subsection
{
Pressure-Energy SPH
}
\subsection
{
Pressure-Energy SPH
}
\label
{
sec:sph:pu
}
\label
{
sec:sph:pu
}
Section 2.2.2 of
\cite
{
Hopkins2013
}
.
\\
\tbd
Section 2.2.2 of
\cite
{
Hopkins2013
}
describes the equations for Pressure-Energy
(P-U) SPH; they are reproduced here with some more details.
P-U SPH depends on the calculation of a smoothed pressure, and follows the
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,
\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,
\\
\label
{
eq:sph:pu:yichoice
}
\end{align}
with the resulting variables representing a smoothed pressure and particle
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
.
\nu
_{
ij
}
\bar
{
\nabla
_
i W
_{
ij
}}
\right
]~.
\label
{
eq:sph:pu:eom
}
\end{align}
which includes the Monaghan artificial viscosity term and Balsara switch in
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
}
\label
{
eq:sph:pu:fij
}
\end{align}
with
$
n
_
d
$
the number of dimensions. In practice, the majority of
$
f
_{
ij
}$
is
precomputed in
{
\tt
hydro
\_
prepare
\_
force
}
as only the curly-bracketed term
depends on the
$
j
$
particle. This cuts out on the majority of operations,
including expensive divisions.
In a similar fashion to
\MinimalSPH
, the internal energy must also be
evolved. Following
\cite
{
Hopkins2013
}
, this is calculated as
\begin{align}
\frac
{
\mathrm
{
d
}
u
_
i
}{
\mathrm
{
d
}
t
}
=
\sum
_
j (
\gamma
- 1)
^
2 m
_
j u
_
j u
_
i
\frac
{
f
_{
ij
}}{
\bar
{
P
}_
i
}
(
\mathbf
{
v
}_
i -
\mathbf
{
v
}_
j)
\cdot
\nabla
_
i W
_{
ij
}
(h
_
i)~.
\label
{
eq:sph:pu:dudt
}
\end{align}
\subsubsection
{
Time integration
}
Time integration follows exactly the same scheme as
\MinimalSPH
.
\subsubsection
{
Particle properties prediction
}
The prediciton of particle properties follows exactly the same scheme as
\MinimalSPH
.
\subsection
{
Anarchy SPH
}
\subsection
{
Anarchy SPH
}
Dalla Vecchia (
\textit
{
in prep.
}
), also described in section 2.2.2 of
Dalla Vecchia (
\textit
{
in prep.
}
), also described in section 2.2.2 of
\cite
{
Schaller2015
}
.
\\
\cite
{
Schaller2015
}
.
\\
...
...
theory/SPH/swift_sph.tex
View file @
c7960649
...
@@ -29,7 +29,7 @@
...
@@ -29,7 +29,7 @@
\section
{
Equation of state
}
\section
{
Equation of state
}
\input
{
EoS/eos
}
\input
{
EoS/eos
}
\section
{
Derivation of the Equation of Motion
}
\section
{
Derivation of the Equation of Motion
}
\label
{
sec:derivation
}
\input
{
Derivation/sph
_
derivation.tex
}
\input
{
Derivation/sph
_
derivation.tex
}
\section
{
SPH flavours
}
\section
{
SPH flavours
}
...
...
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