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0b1efc32
Commit
0b1efc32
authored
Dec 18, 2018
by
Martin Schneider
Browse files
[handbook][box] Replace V_j by B_j
parent
bb148e5b
Changes
1
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Inline
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doc/handbook/5_spatialdiscretizations.tex
View file @
0b1efc32
...
...
@@ 146,16 +146,16 @@ Here, a mass lumping technique is applied by assuming that the storage capacity
reduced to the nodes. This means that the integrals
$
M
_{
i,j
}
=
\int
_{
B
_
j
}
N
_
i
\:
dx
$
are replaced by some mass lumped terms
$
M
^{
lump
}_{
i,j
}$
which are defined as
\begin{equation}
M
^{
lump
}_{
i,j
}
=
\begin{cases}
V
_
j
&
j = i
\\
M
^{
lump
}_{
i,j
}
=
\begin{cases}
B
_
j

&
j = i
\\
0
&
j
\neq
i,
\\
\end{cases}
\end{equation}
where
$
V
_
j
$
is the volume of the FV box
$
B
_
j
$
associated with node
$
j
$
.
where
$
B
_
j

$
is the volume of the FV box
$
B
_
j
$
associated with node
$
j
$
.
The application of this assumption yields
\begin{equation}
\label
{
eq:disc1
}
V
_
j
\frac
{
\partial
\hat
u
_
j
}{
\partial
t
}
B
_
j

\frac
{
\partial
\hat
u
_
j
}{
\partial
t
}
+
\int
_{
\partial
B
_
j
}
F(
\tilde
u)
\cdot
\mathbf
n
\:
d
\varGamma
_{
B
_
j
}
 Q
_
j = 0,
\end{equation}
where
$
Q
_
j
$
is an approximation (using some quadrature rule) of the integrated source/sink term
$
\int
_{
B
_
j
}
q
\:
dx
$
.
...
...
@@ 166,7 +166,7 @@ flow and transport equations:
\begin{equation}
\label
{
eq:discfin
}
V
_
j
\frac
{
\hat
u
_
j
^{
n+1
}

\hat
u
_
j
^{
n
}}{
\Delta
t
}
B
_
j

\frac
{
\hat
u
_
j
^{
n+1
}

\hat
u
_
j
^{
n
}}{
\Delta
t
}
+
\int
_{
\partial
B
_
j
}
F(
\tilde
u
^{
n+1
}
)
\cdot
\mathbf
n
\;
d
{
\varGamma
}_{
B
_
j
}
 Q
_
j
^{
n+1
}
\:
= 0.
\end{equation}
...
...
Timo Koch
@timok
mentioned in commit
a47c325a
·
Dec 18, 2018
mentioned in commit
a47c325a
mentioned in commit a47c325a10eca6ea83d1b420584b0925f9a4de2f
Toggle commit list
Timo Koch
@timok
mentioned in merge request
!1428 (merged)
·
Dec 18, 2018
mentioned in merge request
!1428 (merged)
mentioned in merge request !1428
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