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User:Arthurv~enwiki
Open Source CFD
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Newton iterations:
V
I
n
+
1
(
3
2
Δ
t
+
1
Δ
τ
−
J
m
)
(
U
I
m
+
1
−
U
I
m
)
=
R
(
U
m
)
+
4
(
V
I
U
I
)
n
−
(
V
I
U
I
)
n
−
1
2
Δ
t
−
3
V
I
n
+
1
U
I
m
2
Δ
t
{\displaystyle V_{I}^{n+1}({\frac {3}{2\Delta t}}+{\frac {1}{\Delta \tau }}-J^{m})(U_{I}^{m+1}-U_{I}^{m})=R(U^{m})+{\frac {4(V_{I}U_{I})^{n}-(V_{I}U_{I})^{n-1}}{2\Delta t}}-{\frac {3V_{I}^{n+1}U_{I}^{m}}{2\Delta t}}}
Jacobi iterations:
A
D
Δ
U
I
m
+
1
k
+
1
=
−
∑
i
=
1
N
A
J
i
O
D
Δ
U
J
i
m
+
1
k
+
R
I
n
{\displaystyle A^{D}\Delta U_{I}^{{m+1}^{k+1}}=-\sum _{i=1}^{N}A_{Ji}^{OD}\Delta U_{Ji}^{{m+1}^{k}}+R_{I}^{n}}