Graphics Reference
In-Depth Information
integration since we are approximating the solution
q
(
t
)=
t
0
f
(
q
(
τ
))
dτ,
which has an integral in it.
The simplest time integration method is
forward Euler
, based on the
first one-sided finite difference formula we saw:
q
n
+1
q
n
−
∂q
∂t
(
t
n
)+
O
(Δ
t
)
.
=
Δ
t
Plugging in the differential equation and rearranging gives the formula for
the new value
q
n
+1
based on the previous value
q
n
:
q
n
+1
=
q
n
+Δ
tf
(
q
n
)
This is only first-order accurate, however.
This topic makes use of a few more advanced time integration schemes,
such as Runge-Kutta methods. The Runge-Kutta family gets higher-order
accuracy and other numerical advantages by evaluating
f
at several points
during a time step. For example, one of the classic second-order accurate
Runge-Kutta methods can be written as
q
n
+1
/
2
=
q
n
+
2
Δ
tf
(
q
n
)
,
q
n
+1
=
q
n
+Δ
tf
(
q
n
+1
/
2
)
.
One of the better third-order accurate Runge-Kutta formulas is
k
1
=
f
(
q
n
)
,
k
2
=
f
(
q
n
+
2
Δ
tk
1
)
,
k
3
=
f
(
q
n
+
4
Δ
tk
2
)
,
q
n
+1
=
q
n
+
9
Δ
tk
1
+
9
Δ
tk
2
+
9
Δ
tk
3
.
Note that these are not easy to derive in general!
Many time integration schemes come with a caveat that unless Δ
t
is
chosen small enough, the computed solution exponentially blows up despite
the exact solution staying bounded. This is termed a
stability time-step
restriction
. For some problems, a time integration scheme may even be
unstable no matter how small Δ
t
is: both forward Euler and the second-
order accurate Runge-Kutta scheme above suffer from this flaw in some
cases. The third-order accurate Runge-Kutta scheme may be considered
the simplest general-purpose method as a result.
