6.2 Methods for Initial Value Problems in ODES
137
which is called the trapezoid rule and is the basis for a popular method of
solving partial differential equations - the Crank-Nicolson method.
Collectively, these methods are called two-level methods because they involve the values of the unknown a t only two times (the midpoint rule may or
may not be regarded as a two-level method depending on what further approximations are employed). Analysis of these methods can be found in texts
on the numerical solution of ordinary differential equations (see the bibliography) and will not be repeated here. We shall simply review some of their
most important properties. First we note that all of the methods but the first
require the value of 4(t) a t some point other than t = t, (which is the initial
point of the integration interval at which the solution is known). Therefore,
for these methods, the right hand side cannot be calculated without further
approximation or iteration. Thus, the first method belongs to the class called
explicit methods while all of the others are implicit.
All methods produce good solutions if At is small. However, the behavior
of methods for large step size is important because, in problems with widely
varying time scales (including many problems in fluid mechanics), the goal
is often t o compute the slow, long term behavior of the solution and the
short time scales are merely a nuisance. Problems with a wide range of time
scales are called stiff and are the greatest difficulty one faces in the solution
of ordinary differential equations. It is therefore important t o inquire about
the behavior of methods when the step size is large. This raises the issue of
stability.
There are a number of definitions of stability in the literature. We shall use
a rough definition that calls a method stable if it produces a bounded solution
when the solution of the underlying differential equation is also bounded. For
the explicit Euler method, stability requires:
which, iff (t, 4) is allowed to have complex values, requires that At d f (t, +)/a4
be restricted t o the unit circle with center a t -1. (Complex values must be
considered because higher order systems may have complex eigenvalues. Only
values with zero or negative real part are of interest because they lead to
bounded solutions.) A method with this property is called conditionally stable; for real values of f , Eq. (6.7) reduces to (see Eq. (6.1)):
All of the other methods defined above are unconditionally stable i.e. they
produce bounded solutions for any time step if d f (t, 4)/d+ < 0. However, the
implicit Euler method tends t o produce smooth solutions even when At is
very large while the trapezoid rule frequently yields solutions which oscillate
with little damping. Consequently, the implicit Euler method tends to behave
137
which is called the trapezoid rule and is the basis for a popular method of
solving partial differential equations - the Crank-Nicolson method.
Collectively, these methods are called two-level methods because they involve the values of the unknown a t only two times (the midpoint rule may or
may not be regarded as a two-level method depending on what further approximations are employed). Analysis of these methods can be found in texts
on the numerical solution of ordinary differential equations (see the bibliography) and will not be repeated here. We shall simply review some of their
most important properties. First we note that all of the methods but the first
require the value of 4(t) a t some point other than t = t, (which is the initial
point of the integration interval at which the solution is known). Therefore,
for these methods, the right hand side cannot be calculated without further
approximation or iteration. Thus, the first method belongs to the class called
explicit methods while all of the others are implicit.
All methods produce good solutions if At is small. However, the behavior
of methods for large step size is important because, in problems with widely
varying time scales (including many problems in fluid mechanics), the goal
is often t o compute the slow, long term behavior of the solution and the
short time scales are merely a nuisance. Problems with a wide range of time
scales are called stiff and are the greatest difficulty one faces in the solution
of ordinary differential equations. It is therefore important t o inquire about
the behavior of methods when the step size is large. This raises the issue of
stability.
There are a number of definitions of stability in the literature. We shall use
a rough definition that calls a method stable if it produces a bounded solution
when the solution of the underlying differential equation is also bounded. For
the explicit Euler method, stability requires:
which, iff (t, 4) is allowed to have complex values, requires that At d f (t, +)/a4
be restricted t o the unit circle with center a t -1. (Complex values must be
considered because higher order systems may have complex eigenvalues. Only
values with zero or negative real part are of interest because they lead to
bounded solutions.) A method with this property is called conditionally stable; for real values of f , Eq. (6.7) reduces to (see Eq. (6.1)):
All of the other methods defined above are unconditionally stable i.e. they
produce bounded solutions for any time step if d f (t, 4)/d+ < 0. However, the
implicit Euler method tends t o produce smooth solutions even when At is
very large while the trapezoid rule frequently yields solutions which oscillate
with little damping. Consequently, the implicit Euler method tends to behave