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CHAPTER 9. INITIAL VALUE PROBLEMS FOR ODES
Source
FMM
HSL
IMSL
KMN
MATLAB
NAG
netlib
NR
NUMAL
ODEPACK
SLATEC
TOMS
TOMS
9.8
Table 9.2: Software for ODE initial value problems
Runge-Kutta
Adams
Stiff
rkf45
da02
dc03
ivprk
ivpag
ivpag
sdriv2
sdriv2
ode23/ode45
ode113
ode15s/ode23s
d02baf
d02caf
d02eaf
dverk
ode
vode/vodpk
odeint
stiff
rke
multistep
gms
lsode
lsode
derkf
deabm/sdriv1 debdf/sdriv2
gerk(#504)
stint(#534)
brk45(#669)/rkn(#670)
mebdf(#703)
Historical Notes and Further Reading
Euler proposed his method for initial value problems in 1768. Much of the early impetus
for the numerical solution of ordinary differential equations was from celestial mechanics.
For example, in 1846 Adams—for whom the classical linear multistep methods are named—
finished in a dead heat with Le Verrier in accurately predicting the location at which the
planet Neptune would be discovered. Their orbital calculations were based on known but
previously unexplained perturbations in the orbit of Uranus.
Runge-Kutta methods were developed independently by Runge and Kutta around 1900.
Fehlberg’s implementation, which permitted an efficient error estimate, was developed in
the 1960s. A practical method based on extrapolation was published by Bulirsch and Stoer
in 1966. Gear’s method for solving stiff ODEs was published in 1971, along with a very
influential computer program difsub (TOMS #407) implementing the method. Another
influential code for solving ODEs was developed at about the same time by Krogh. Multivalue methods were first proposed by Nordsieck in 1962 to address the implementation
difficulties of multistep methods. For the equivalence of multistep and multivalue methods,
see Skeel [230].
Recent books on the numerical solution of initial value problems for ODEs include [66,
133, 156, 224]. Earlier textbooks and monographs on this topic include [31, 77, 91, 123,
155, 160, 227]. In addition, see the surveys [32, 58, 92, 111, 226, 228, 238]. Practical advice
on using ODE software can be found in [223]. For solving differential-algebraic systems,
see [22].
Review Questions
9.1 True or false: An ODE whose solution
curves are unbounded as time increases is necessarily unstable.
9.2 True or false: In the numerical solution
of an ODE, the global error grows only if the
equation is unstable.
REVIEW QUESTIONS
9.3 True or false: In solving an ODE numerically, the roundoff error and the truncation
error are independent of each other.
9.4 True or false: In numerically solving an
initial value problem for an ODE, the global
truncation error is always at least as large as
the sum of the local truncation errors.
9.5 True or false: For solving a stable
differential equation numerically, an implicit
method is always stable.
9.6 True or false: With an unconditionally
stable method, one can take arbitrarily large
time steps in numerically solving a stable ODE
to achieve a given accuracy.
9.7 True or false: Stiff ODEs are always difficult and expensive to solve.
9.8 (a) In general, does a differential equation, by itself, determine a unique solution?
(b) If so, why, and if not, what additional information must be specified to determine a solution uniquely?
9.9 (a) What is meant by a first-order ODE?
(b) Why are higher-order ODEs usually transformed into equivalent first-order ODEs before
solving them numerically?
9.10 (a) Describe in words the distinction between a stable ODE and an unstable ODE.
(b) What is a mathematical criterion for determining the stability of a system of ODEs
y 0 = f (t, y)?
(c) Can the stability or instability of a system
of ODEs change with time?
9.11 Which of the following types of firstorder ODEs are stable?
(a) An equation whose solution curves converge toward each other
(b) An equation whose Jacobian is negative
(c) A stiff equation
(d ) An equation with exponentially decaying
solutions
9.12 Classify each of the following ODEs as
stable, unstable, or neutrally stable.
(a) y 0 = y + t.
(b) y 0 = y − t.
301
(c) y 0 = t − y.
(d ) y 0 = 1.
9.13 How does a typical numerical solution of
an ODE differ from an analytical solution?
9.14 (a) What is Euler’s method for solving
an ODE?
(b) Show how it is derived.
9.15 Describe in words the difference between
the local truncation error and the global truncation error in solving an initial value problem
for an ODE numerically.
9.16 Under what condition is the global error
in solving an initial value problem for an ODE
likely to be smaller than the sum of the local
errors at each step?
9.17 In solving an ODE numerically, which
is usually more significant, rounding error or
truncation error?
9.18 (a) Define in words the error amplification factor for one step of a numerical method
for solving an initial value problem for an
ODE.
(b) Does the amplification factor depend only
on the equation, only on the method of solution, or on both?
(c) What is the value of the amplification factor for one step of Euler’s method?
(d ) What stability region does this imply for
Euler’s method?
9.19 (a) What is the basic difference between
an explicit method and an implicit method for
solving an ODE numerically?
(b) Comparing these two types of methods, list
one relative advantage for each.
(c) Name a specific example of a method (or
family of methods) of each type.
9.20 The use of an implicit method for solving a nonlinear ODE requires the iterative solution of a nonlinear equation. How can one
get a good starting guess for this iteration?
9.21 Is it possible for a numerical solution
method to be unstable when applied to a stable ODE?
9.22 What does it mean for a numerical ODE
method to be of order p?
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CHAPTER 9. INITIAL VALUE PROBLEMS FOR ODES
9.23 (a) For solving ODEs, what is the
highest-order accuracy that a linear multistep
method can have and still be unconditionally
stable?
(b) Give an example of a method having these
properties (by name or by formula).
9.24 Compare the stability regions (i.e., the
stability constraints on the stepsize) for the
Euler and backward Euler methods for solving
a scalar ODE.
9.25 For the backward Euler method, which
factor places a stronger restriction on the
choice of stepsize: stability or accuracy?
9.26 Which of the following numerical methods for solving a stable ODE numerically are
unconditionally stable?
(a) Euler’s method
(b) Backward Euler method
(c) Trapezoid rule
9.27 (a) What is meant by a stiff ODE?
(b) Why may a stiff ODE be difficult to solve
numerically?
9.33 (a) What is the basic difference between
a single-step method and a multistep method
for solving an ODE numerically?
(b) Comparing these two types of methods, list
one relative advantage for each.
(c) Name a specific example of a method (or
family of methods) of each type.
9.34 List two advantages and two disadvantages of multistep methods compared with
classical Runge-Kutta methods for solving
ODEs numerically.
9.35 What is the principal drawback of a Taylor series method compared with a RungeKutta method for solving ODEs?
9.36 (a) What is the principal advantage of
extrapolation methods for solving ODEs numerically?
(b) What are the disadvantages of such methods?
9.37 In using a multistep method to solve an
ODE numerically, why might one still need to
have a single-step method available?
(c) What type of method is appropriate for
solving stiff ODEs?
9.38 Why are multistep methods for solving ODEs numerically often used in predictorcorrector pairs?
9.28 Suppose one is using the backward Euler method to solve a nonlinear ODE numerically. The resulting nonlinear algebraic equation at each step must be solved iteratively.
If a fixed number of iterations are performed
at each step, is the resulting method unconditionally stable?
9.39 If a predictor-corrector method for solving an ODE is implemented as a PECE
scheme, does the second evaluation affect the
value obtained for the solution at the point being computed? If so, what is the effect, and if
not, then why is the second evaluation done?
9.29 Explain why implicit methods are better
than explicit methods for solving stiff systems
of ODEs numerically.
9.40 List two reasons why multivalue methods are easier to implement than multistep
methods for solving ODEs adaptively with automatic error control.
9.30 What is the simplest numerical method
that is stable for integrating a stiff ODE?
9.31 For solving ODEs numerically, why is
it usually impractical to generate methods of
very high accuracy by using many terms in a
Taylor series expansion?
9.32 In solving an ODE numerically, with
which type of method, Runge-Kutta or multistep, is it easier to supply values for the numerical solution at arbitrary output points within
each step?
9.41 For each of the following properties,
state which type of ODE method, multistep
or classical Runge-Kutta, more accurately fits
the description:
(a) Self starting
(b) More efficient in attaining high accuracy
(c) Can be efficient for stiff problems
(d ) Easier to program
(e) Easier to change stepsize
(f ) Easier to obtain a local error estimate
EXERCISES
303
(g) Easier to produce output at arbitrary intermediate points within each step
9.42 Give two approaches to starting a multistep method initially when past solution history is not yet available.
Exercises
9.1 Write each of the following ODEs as an
equivalent first-order system of ODEs:
(a) y 00 = t + y + y 0 .
(b) y 000 = y 00 + ty.
(c) y 000 = y 00 − 2y 0 + y − t + 1.
9.2 Write each of the following ODEs as an
equivalent first-order system of ODEs:
(a) Van der Pol equation:
00
0
2
y = y (1 − y ) − y.
(b) Blasius equation:
y 000 = −y y 00 .
(c) Newton’s Second Law of Motion for twobody problem:
y100
y200
= −GM y1 /(y12 + y22 )3/2 ,
= −GM y2 /(y12 + y22 )3/2 .
9.3 Is the following system of ODEs stable?
y10
y20
= −y1 + y2 ,
= −2y2 .
Explain your answer.
9.4 Consider the ODE y 0 = −5y with initial
condition y(0) = 1. We will solve this ODE
numerically using a stepsize of h = 0.5.
(a) Is this ODE stable?
(b) Is Euler’s method stable for this ODE using this stepsize?
(c) Compute the numerical value for the approximate solution at t = 0.5 given by Euler’s
method.
(d ) Is the backward Euler method stable for
this ODE using this stepsize?
(e) Compute the numerical value for the approximate solution at t = 0.5 given by the
backward Euler method.
9.5 With an initial value of y0 = 1 at t0 = 0
and a time step of h = 1, compute the approximate solution value y1 at time t1 = 1 for the
ODE y 0 = −y using each of the following two
numerical methods. (Your answers should be
numbers, not formulas.)
(a) Euler’s method
(b) Backward Euler method
9.6 For the ODE, initial value, and stepsize
given in Example 9.10, prove that fixed-point
iteration for solving the implicit equation for
y1 is in fact convergent. What is the convergence rate?
9.7 Consider the initial value problem
y 00 = y
for t ≥ 0, with initial values y(0) = 1 and
y 0 (0) = 2.
(a) Express this second-order ODE as an
equivalent system of two first-order ODEs.
(b) What are the corresponding initial conditions for the system of ODEs in part a?
(c) Is this a stable system of ODEs?
(d ) Perform one step of Euler’s method for this
ODE system using a stepsize of h = 0.5.
(e) Is Euler’s method stable for this problem
using this stepsize?
(f ) Is the backward Euler method stable for
this problem using this stepsize?
9.8 Consider the initial value problem for
the ODE y 0 = −y 2 with the initial condition
y(0) = 1. We will use the backward Euler
method to compute the approximate value of
the solution y1 at time t1 = 0.1 (i.e., take
one step using the backward Euler method
with stepsize h = 0.1 starting from y0 = 1
at t0 = 0). Since the backward Euler method
is implicit, and the ODE is nonlinear, we will
need to solve a nonlinear algebraic equation for
y1 .
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CHAPTER 9. INITIAL VALUE PROBLEMS FOR ODES
(a) Write out that nonlinear algebraic equation for y1 .
(b) Write out the Newton iteration for solving
the nonlinear algebraic equation.
(c) Obtain a starting guess for the Newton iteration by using one step of Euler’s method for
the ODE.
(d ) Finally, compute an approximate value for
the solution y1 by using one iteration of Newton’s method for the nonlinear algebraic equation.
(c) Self-starting
9.9 For solving an ODE y 0 = f (t, y) numerically, each of the following methods,
(1)
9.11 The centered difference approximation
yk+1
1
= yk + [f (tk , yk )
2
+f (tk+1 , yk + f (tk , yk )h)]h,
(2)
yk+1
1
= yk + [3f (tk , yk )
2
−f (tk−1 , yk−1 )]h,
(3)
yk+1
1
= yk + [f (tk , yk )
2
+f (tk+1 , yk+1 )]h
is of second order, but the methods also
have important differences. For each property
listed, state which of the three methods has or
have the given property.
(a) Single-step method
(b) Implicit method
(d ) Unconditionally stable
(e) Runge-Kutta type method
(f ) Good for solving a stiff ODE
9.10 Use the linear ODE y 0 = λy to analyze
the accuracy and stability of Heun’s method
(see Section 9.6.2). In particular, verify that
this method is second-order accurate, and describe or plot its stability region in the complex
plane.
y0 ≈
yk+1 − yk−1
2h
leads to the two-step leapfrog method
yk+1 = yk−1 + f (tk , yk )2h
for solving the ODE y 0 = f (t, y). Determine
the order of accuracy and the stability region
of this method.
9.12 Let A be an n × n matrix. Compare and
contrast the behavior of the linear difference
equation
xk+1 = Axk
with that of the linear differential equation
x0 = Ax.
What is the general solution in each case?
In each case, what property of the matrix
A would imply that the solution remains
bounded for any starting vector x0 ? You may
assume that the matrix A is diagonalizable.
Computer Problems
9.1 The populations of two species, a prey
denoted by y1 and predator denoted by y2 , can
be modeled by a system of ODEs
y10
y20
= by1 − cy1 y2 ,
= −dy2 + cy1 y2
due to Lotka and Volterra. The parameters b
and d govern the birth rate of prey and death
rate of predators, respectively, and the param-
eter c governs the interaction of the two populations. With the parameter values b = 1,
d = 10, and c = 1, and initial conditions
y1 (0) = 0.5 and y2 (0) = 1 (the populations
are normalized, and we treat them as continuous variables), use a library routine to solve
this system numerically, integrating to t = 10.
Plot each of the two populations as a function
of time, and on a separate graph plot the tra-