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version 4.2
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Oct 29, 2025
by
Udo Ziegler
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@@ -965,15 +965,11 @@ integration over cell faces turns out to be too complicated.
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@@ -965,15 +965,11 @@ integration over cell faces turns out to be too complicated.
The face-averaged magnetic field components are then obtained from the
The face-averaged magnetic field components are then obtained from the
integral form of $
\m
athbf{B}=
\n
abla
\t
imes
\m
athbf{A}$:
integral form of $
\m
athbf{B}=
\n
abla
\t
imes
\m
athbf{A}$:
`g->bx[iz][iy][ix]`
=
\[
*h*
<sub>
*y*
</sub>
*Δ*
<sub>
`
iy
`
</sub>
(
*h*
<sub>
*zy*
</sub>
*Â*
<sub>
*z*
</sub>
)−
*h*
<sub>
*y*
</sub>
*Δ*
<sub>
`
iz
`
</sub>
*Â*
<sub>
*y*
</sub>
\]
/
*δ*
𝒜
<sub>
*x*
</sub>
`g->bx[iz][iy][ix]`
=
\[
*h*
<sub>
*y*
</sub>
*Δ*
<sub>
iy
</sub>
(
*h*
<sub>
*zy*
</sub>
*Â*
<sub>
*z*
</sub>
)−
*h*
<sub>
*y*
</sub>
*Δ*
<sub>
iz
</sub>
*Â*
<sub>
*y*
</sub>
\]
/
*δ*
𝒜
<sub>
*x*
</sub>
`g->by[iz][iy][ix]`
=
\[
*Δ*
<sub>
`iz`
</sub>
*Â*
<sub>
*x*
</sub>
−
*h*
<sub>
*zy*
</sub>
*Δ*
<sub>
`ix`
</sub>
(
*h*
<sub>
*y*
</sub>
*Â*
<sub>
*z*
</sub>
)
\]
/
*δ*
𝒜
<sub>
*y*
</sub>
`g->by[iz][iy][ix]`
=
\[
*Δ*
<sub>
iz
</sub>
*Â*
<sub>
*x*
</sub>
−
*h*
<sub>
*zy*
</sub>
*Δ*
<sub>
ix
</sub>
(
*h*
<sub>
*y*
</sub>
*Â*
<sub>
*z*
</sub>
)
\]
/
*δ*
𝒜
<sub>
*y*
</sub>
`g->bz[iz][iy][ix]`
=
\[
*Δ*
<sub>
`ix`
</sub>
(
*h*
<sub>
*y*
</sub>
*Â*
<sub>
*y*
</sub>
)−
*Δ*
<sub>
`iy`
</sub>
*Â*
<sub>
*x*
</sub>
\]
/
*δ*
𝒜
<sub>
*z*
</sub>
where (
*Â*
<sub>
*x*
</sub>
,
*Â*
<sub>
*y*
</sub>
,
*Â*
<sub>
*z*
</sub>
) denote
the path integrals
`g->bz[iz][iy][ix]`
=
\[
*Δ*
<sub>
ix
</sub>
(
*h*
<sub>
*y*
</sub>
*Â*
<sub>
*y*
</sub>
)−
*Δ*
<sub>
iy
</sub>
*Â*
<sub>
*x*
</sub>
\]
/
*δ*
𝒜
<sub>
*z*
</sub>
`g->bx[iz][iy][ix]`
$=
\l
eft[h_y
\D
elta_{iy} (h_{zy}
\h
at{A}_z)
`g->bx[iz][iy][ix]`
$=
\l
eft[h_y
\D
elta_{iy} (h_{zy}
\h
at{A}_z)
-h_y
\D
elta_{iz}
\h
at{A}_y
\r
ight]/
\d
elta
\m
athcal{A}_x$
-h_y
\D
elta_{iz}
\h
at{A}_y
\r
ight]/
\d
elta
\m
athcal{A}_x$
...
@@ -994,6 +990,8 @@ $\Delta_{iy}X=X(\mathtt{iz,iy+1,ix})-X(\mathtt{iz,iy,ix})$,
...
@@ -994,6 +990,8 @@ $\Delta_{iy}X=X(\mathtt{iz,iy+1,ix})-X(\mathtt{iz,iy,ix})$,
$
\D
elta_{iz}X=X(
\m
athtt{iz,iy,ix+1})-X(
\m
athtt{iz,iy,ix})$
$
\D
elta_{iz}X=X(
\m
athtt{iz,iy,ix+1})-X(
\m
athtt{iz,iy,ix})$
and $(
\h
at{A}_x,
\h
at{A}_y,
\h
at{A}_z)$ are the cell-edge integrals
and $(
\h
at{A}_x,
\h
at{A}_y,
\h
at{A}_z)$ are the cell-edge integrals
and (
*Â*
<sub>
*x*
</sub>
,
*Â*
<sub>
*y*
</sub>
,
*Â*
<sub>
*z*
</sub>
)
are the cell-edge integrals
...
...
...
...
