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In-Depth Information
D
.
u
nC1
;
v
nC1
/
T
We observe that in order to compute
w
nC1
based on
w
n
D
.
u
n
;
v
n
/, we have to solve the linear system (
4.133
). Since
det.
A
/
D
1
C
t
2
>0;
(4.135)
we know that (
4.133
) has a unique solution given by
w
nC1
D
A
1
w
n
:
(4.136)
Here
1
t
t
;
1
1
C
t
2
A
1
D
(4.137)
1
see (3.63) on page 85. From (
4.136
)and(
4.137
)wehave
u
nC1
v
nC1
D
1
t
t
u
n
v
n
D
u
n
t
v
n
t
u
n
C
v
n
; (4.138)
1
1
C
t
2
1
1
C
t
2
1
or
1
1
C
t
2
u
nC1
D
.
u
n
t
v
n
/;
1
1
C
t
2
v
nC1
D
.
v
n
C
t
u
n
/:
(4.139)
By choosing
u
0
D
1 and
v
0
D
0, we have the analytical solutions
u
.t /
D
cos.t /;
v
.t /
D
sin.t /:
(4.140)
In Fig.
4.8
we have plotted .
u
;
v
/ and .
u
n
;
v
n
/ for 0
t
2, t
D
=500.We
observe that the scheme provides good approximations.
4.6.2
A Nonlinear System
We saw that the fully implicit discretization of the linear system of ODEs (
4.129
)
leads to a linear algebraic system of equations, see (
4.133
). Let us now consider the
following nonlinear system of ODEs:
u
0
D
v
3
;
u
.0/
D
u
0
;
(4.141)
v
0
D
u
3
;
v
.0/
D
v
0
:
