Processing math: 100%
A centered scheme for linear damping
[
m
D
t
D
t
u
+
f
(
D
2
t
u
)
+
s
(
u
)
=
F
]
n
Written out
m
u
n
+
1
−
2
u
n
+
u
n
−
1
Δ
t
2
+
f
(
u
n
+
1
−
u
n
−
1
2
Δ
t
)
+
s
(
u
n
)
=
F
n
Assume
f
(
u
′
)
is linear in
u
′
=
v
:
u
n
+
1
=
(
2
m
u
n
+
(
b
2
Δ
t
−
m
)
u
n
−
1
+
Δ
t
2
(
F
n
−
s
(
u
n
)
)
)
(
m
+
b
2
Δ
t
)
−
1
