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6. Methods for Unsteady Problems
A common method uses the (m - 1)st order Adams-Bashforth method as a
predictor and the m t h order Adams-Moulton method as a corrector. Thus,
predictor-corrector methods of any order can be obtained.
The multipoint approach has the advantage that it is relatively easy to
construct methods of any order. These methods are also easy t o use and
program. A final advantage is that they require only one evaluation of the
derivative per time step (which may be very complicated, especially in applications involving partial differential equations), making them relatively
cheap. (The values of f (t,4(t)) are required several times, but they can be
stored once calculated, so that only one evaluation per time step is required.)
Their principal disadvantage is that, because they require data from many
points prior to the current one, they cannot be started using only data a t the
initial time point. One has t o use other methods to get calculation started.
One approach is to use a small step size and a lower order method (so that
the desired accuracy is achieved) and slowly increase the order as more data
become available.
These methods are the basis for many accurate ordinary differential equation solvers. In these solvers, error estimators are used to determine the accuracy of the solution a t every step. If the solution is not accurate enough,
the order of the method is increased up to the maximum order allowed by
the program. On the other hand, if the solution is much more accurate than
necessary, the order of the method might be reduced to save computing time.
Because the step size is difficult to change in multipoint methods, this is done
only when the maximum order of the method has already been reached.'
Because multipoint methods use data from several time steps, they may
produce non-physical solutions. Space does not permit inclusion of the analysis here but it is worth noting that the instabilities of multipoint methods
are most often due to the non-physical solutions. These may be suppressed,
but not entirely, by a careful choice of the starting method. It is common
for these methods to give an accurate solution for some time and then begin to behave badly as the non-physical component of the solution grows. A
common remedy for this problem is to restart the method every so often, a
trick that is effective but may reduce the accuracy and/or the efficiency of
the scheme.
6.2.3 Runge-Kutta Methods
The difficulties in starting multipoint methods can be overcome by using
points between t, and t,+l rather than earlier points. Methods of this kind
are called Runge-Kutta methods. We shall give just two examples.
The quadrature approximations used above assume a uniform time step; if the
time step is allowed to vary, the coefficients multiplying the function values at
different time levels become complicated functions of step sizes, as we saw in
Chap. 3 for finite differences in space.
6. Methods for Unsteady Problems
A common method uses the (m - 1)st order Adams-Bashforth method as a
predictor and the m t h order Adams-Moulton method as a corrector. Thus,
predictor-corrector methods of any order can be obtained.
The multipoint approach has the advantage that it is relatively easy to
construct methods of any order. These methods are also easy t o use and
program. A final advantage is that they require only one evaluation of the
derivative per time step (which may be very complicated, especially in applications involving partial differential equations), making them relatively
cheap. (The values of f (t,4(t)) are required several times, but they can be
stored once calculated, so that only one evaluation per time step is required.)
Their principal disadvantage is that, because they require data from many
points prior to the current one, they cannot be started using only data a t the
initial time point. One has t o use other methods to get calculation started.
One approach is to use a small step size and a lower order method (so that
the desired accuracy is achieved) and slowly increase the order as more data
become available.
These methods are the basis for many accurate ordinary differential equation solvers. In these solvers, error estimators are used to determine the accuracy of the solution a t every step. If the solution is not accurate enough,
the order of the method is increased up to the maximum order allowed by
the program. On the other hand, if the solution is much more accurate than
necessary, the order of the method might be reduced to save computing time.
Because the step size is difficult to change in multipoint methods, this is done
only when the maximum order of the method has already been reached.'
Because multipoint methods use data from several time steps, they may
produce non-physical solutions. Space does not permit inclusion of the analysis here but it is worth noting that the instabilities of multipoint methods
are most often due to the non-physical solutions. These may be suppressed,
but not entirely, by a careful choice of the starting method. It is common
for these methods to give an accurate solution for some time and then begin to behave badly as the non-physical component of the solution grows. A
common remedy for this problem is to restart the method every so often, a
trick that is effective but may reduce the accuracy and/or the efficiency of
the scheme.
6.2.3 Runge-Kutta Methods
The difficulties in starting multipoint methods can be overcome by using
points between t, and t,+l rather than earlier points. Methods of this kind
are called Runge-Kutta methods. We shall give just two examples.
The quadrature approximations used above assume a uniform time step; if the
time step is allowed to vary, the coefficients multiplying the function values at
different time levels become complicated functions of step sizes, as we saw in
Chap. 3 for finite differences in space.