36
2 Preliminaries
a)
t
x(t)
t
n−1
t
n
x
n−1
x
n
b)
t
x(t)
t
n−1
t
n
x
n−1
x
n
t
(t)
t
n−1
t
n
n−1
n
d)
t
x(t)
t
n−1
t
n−0.5
t
n
x
n−1
x
n−0.5
x
n
c)
Fig. 2.7 Integrators for ˙
x = f
x(t)
(note that f
x(t)
denotes the slope of x = x(t): a Eulerforward (explicit, conditionally stable, first-order accurate), b Euler-backward (implicit, unconditionally stable, first-order accurate), c Trapezoidal (implicit, unconditionally stable, second-order
accurate), d Midpoint (implicit, unconditionally stable, second-order accurate)
The Euler Backward Integrator
x
n
− x
n−1
=
n f (x
n
).
(2.36)
Implicit, unconditionally stable, and first-order accurate.
The Heun Trapezoidal Integrator
x n − x n−1 = n 1
2
[ f (x p ) + f (x n−1 )] with x p − x n−1 = n f (x n−1 ). (2.37)
Explicit, conditionally stable, and second-order accurate.
The Heun Midpoint Integrator
x
n
− x
n−1
=
n f
1
2
[x
p
+ x
n−1
]
with x
p
− x
n−1
=
n f (x
n−1
). (2.38)
Explicit, conditionally stable, and second-order accurate.