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eb9ea429
Commit
eb9ea429
authored
Dec 11, 2018
by
Martin Schneider
2
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[handbook][disc] Correction of indices used for the box method
parent
9541a0b2
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doc/handbook/5_spatialdiscretizations.tex
doc/handbook/5_spatialdiscretizations.tex
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doc/handbook/5_spatialdiscretizations.tex
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eb9ea429
...
...
@@ 146,19 +146,19 @@ 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
_
i
&
i = j
\\
0
&
i
\neq
j.
\\
M
^{
lump
}_{
i,j
}
=
\begin{cases}
V
_
j
&
j = i
\\
0
&
j
\neq
i,
\\
\end{cases}
\end{equation}
where
$
V
_
i
$
is the volume of the FV box
$
B
_
i
$
associated with node
$
i
$
.
where
$
V
_
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
_
i
\frac
{
\partial
\hat
u
_
i
}{
\partial
t
}
+
\int
_{
\partial
B
_
j
}
F(
\tilde
u)
\cdot
\mathbf
n
\:
d
\varGamma
_{
B
_
j
}
 Q
_
i
= 0,
V
_
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
_
i
$
is an approximation (using some quadrature rule) of the integrated source/sink term
$
\int
_{
B
_
j
}
q
\:
dx
$
.
where
$
Q
_
j
$
is an approximation (using some quadrature rule) of the integrated source/sink term
$
\int
_{
B
_
j
}
q
\:
dx
$
.
Using an implicit Euler time discretization finally
leads to the discretized form which will be applied to the mathematical
...
...
@@ 166,10 +166,11 @@ flow and transport equations:
\begin{equation}
\label
{
eq:discfin
}
V
_
i
\frac
{
\hat
u
_
i
^{
n+1
}

\hat
u
_
i
^{
n
}}{
\Delta
t
}
+
\int
_{
\partial
B
_
i
}
F(
\tilde
u
^{
n+1
}
)
\cdot
\mathbf
n
\;
d
{
\varGamma
}_{
B
_
i
}
 Q
_
i
^{
n+1
}
\:
= 0.
V
_
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}
Equation
\eqref
{
eq:discfin
}
has to be fulfilled for each box
$
B
_
j
$
.
\subsection
{
Cell Centered Finite Volume Method  A Short Introduction
}
\label
{
cc
}
...
...
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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