Information Technology Reference
In-Depth Information
Chapter 4
Nonlinear Algebraic Equations
Suppose we want to solve the ordinary differential equation (ODE)
u
0
D
g.
u
/;
u
.0/
D
u
0
;
(4.1)
by an implicit scheme. The standard implicit Euler method for (
4.1
), cf. Sect. 2.2.4,
reads
u
nC1
t g.
u
nC1
/
D
u
n
;
(4.2)
where
u
n
u
.t
n
/ and t
n
D
nt for a small time step t > 0. We note that in order
to compute
u
nC1
based on
u
n
,wehavetosolve(
4.2
). Let us look a bit closer at this.
For simplicity we let c denote
u
n
and
v
denote
u
nC1
, so we need to solve
v
t g.
v
/
D
c;
(4.3)
where c is given and we want to find
v
.
Let us first consider the case of
g.
u
/
D
u
;
(4.4)
which corresponds to the ODE
u
0
D
u
;
u
.0/
D
u
0
:
(4.5)
Then (
4.3
) can be rewritten as
v
t
v
D
c;
(4.6)
which has the solution
1
1
t
v
D
c:
(4.7)
