138
6. Methods for Unsteady Problems
well, even for non-linear equations, while the trapezoid rule may be unstable
for non-linear problems.
Finally, the question of accuracy needs to be considered. On a global
scale, it is difficult to say much about this issue due to the incredible variety
of equations that one might need to deal with. For a single small step it is
possible to use Taylor series to show that the explicit Euler method, starting
with the known solution at t,, yields the solution at time t,+At with an error
proportional to (At)2. However, as the number of steps required to compute
to some finite final time t = to + T is inversely proportional to At, and an
error is incurred on each step, the error will at the end be proportional t o At
itself. Therefore, explicit Euler method is a first order method. The implicit
Euler method is also a first order method, while the trapezoid and midpoint
rule methods have errors proportional to (At)2 and are therefore second order
methods. It can be shown that second order is the highest order achievable
by a two-level scheme.
It is important t o note that the order of a method is not the sole indicator
of its accuracy. While it is true that, for small enough step size, a high order
method will have a smaller error than a low order one, it is also true that two
methods of the same order may have errors that differ by as much as an order
of magnitude. The order determines only the rate at which the error goes to
zero as the step size goes to zero, and this only after the step size has become
small enough. 'Small enough' is both method and problem dependent and
cannot be determined a priori.
When the step size is small enough, one can estimate the discretization
error in the solution by comparing solutions obtained using two different step
sizes. This method, known as Richardson extrapolation, has been described in
Chap. 3, and applies to both spatial and temporal discretization errors. The
error can also be estimated by analyzing the difference in solutions produced
by two schemes of different order; this will be discussed in Chap. 11.
6.2.2 Predictor-Corrector and Multipoint Methods
The properties that we have found for two-level methods are quite general.
Explicit methods are very easy to program and use little computer memory
and computation time per step but are unstable if the time step is large.
On the other hand, implicit methods require iterative solution to obtain the
values a t the new time step. This makes them harder to program and they
use more computer memory and time per time step, but they are much more
stable. (The implicit methods described above are unconditionally stable; this
is not true of all implicit methods but they are generally more stable than
their explicit counterparts.) One might ask whether it is possible to combine
the best of the two methods. Predictor-corrector methods are an attempt t o
do this.
A wide variety of predictor-corrector methods has been developed; for the
present, we shall give just one, which is so well-known that it is often called
6. Methods for Unsteady Problems
well, even for non-linear equations, while the trapezoid rule may be unstable
for non-linear problems.
Finally, the question of accuracy needs to be considered. On a global
scale, it is difficult to say much about this issue due to the incredible variety
of equations that one might need to deal with. For a single small step it is
possible to use Taylor series to show that the explicit Euler method, starting
with the known solution at t,, yields the solution at time t,+At with an error
proportional to (At)2. However, as the number of steps required to compute
to some finite final time t = to + T is inversely proportional to At, and an
error is incurred on each step, the error will at the end be proportional t o At
itself. Therefore, explicit Euler method is a first order method. The implicit
Euler method is also a first order method, while the trapezoid and midpoint
rule methods have errors proportional to (At)2 and are therefore second order
methods. It can be shown that second order is the highest order achievable
by a two-level scheme.
It is important t o note that the order of a method is not the sole indicator
of its accuracy. While it is true that, for small enough step size, a high order
method will have a smaller error than a low order one, it is also true that two
methods of the same order may have errors that differ by as much as an order
of magnitude. The order determines only the rate at which the error goes to
zero as the step size goes to zero, and this only after the step size has become
small enough. 'Small enough' is both method and problem dependent and
cannot be determined a priori.
When the step size is small enough, one can estimate the discretization
error in the solution by comparing solutions obtained using two different step
sizes. This method, known as Richardson extrapolation, has been described in
Chap. 3, and applies to both spatial and temporal discretization errors. The
error can also be estimated by analyzing the difference in solutions produced
by two schemes of different order; this will be discussed in Chap. 11.
6.2.2 Predictor-Corrector and Multipoint Methods
The properties that we have found for two-level methods are quite general.
Explicit methods are very easy to program and use little computer memory
and computation time per step but are unstable if the time step is large.
On the other hand, implicit methods require iterative solution to obtain the
values a t the new time step. This makes them harder to program and they
use more computer memory and time per time step, but they are much more
stable. (The implicit methods described above are unconditionally stable; this
is not true of all implicit methods but they are generally more stable than
their explicit counterparts.) One might ask whether it is possible to combine
the best of the two methods. Predictor-corrector methods are an attempt t o
do this.
A wide variety of predictor-corrector methods has been developed; for the
present, we shall give just one, which is so well-known that it is often called