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6. TIME-MARCHING METHODS FOR
ODE’S
After discretizing the spatial derivatives in the governing PDE’s (such as the
Navier-Stokes equations), we obtain a coupled system of nonlinear ODE’s in
the form
d~u
= F~ (~u, t)
dt
(6.1)
These can be integrated in time using a time-marching method to obtain a
time-accurate solution to an unsteady flow problem. For a steady flow problem, spatial discretization leads to a coupled system of nonlinear algebraic
equations in the form
F~ (~u) = 0
(6.2)
As a result of the nonlinearity of these equations, some sort of iterative
method is required to obtain a solution. For example, one can consider the use
of Newton’s method, which is widely used for nonlinear algebraic equations
(see Section 6.10.3). This produces an iterative method in which a coupled
system of linear algebraic equations must be solved at each iteration. These
can be solved iteratively using relaxation methods, which will be discussed in
Chapter 9, or directly using Gaussian elimination or some variation thereof.
Alternatively, one can consider a time-dependent path to the steady state
and use a time-marching method to integrate the unsteady form of the equations until the solution is sufficiently close to the steady solution. The subject
of the present chapter, time-marching methods for ODE’s, is thus relevant
to both steady and unsteady flow problems. When using a time-marching
method to compute steady flows, the goal is simply to remove the transient
portion of the solution as quickly as possible; time-accuracy is not required.
This motivates the study of stability and stiffness, topics which are covered
in the next two chapters.
Application of a spatial discretization to a PDE produces a coupled system
of ODE’s. Application of a time-marching method to an ODE produces an
ordinary difference equation (O∆E ). In earlier chapters, we developed exact
solutions to our model PDE’s and ODE’s. In this chapter we will present some
basic theory of linear O∆E’s which closely parallels that for linear ODE’s,
and, using this theory, we will develop exact solutions for the model O∆E’s
arising from the application of time-marching methods to the model ODE’s.
82
6. TIME-MARCHING METHODS FOR ODE’S
6.1 Notation
Using the semi-discrete approach, we reduce our PDE to a set of coupled
ODE’s represented in general by Eq. 4.1. However, for the purpose of this
chapter, we need only consider the scalar case
du
= u0 = F (u, t)
dt
(6.3)
Although we use u to represent the dependent variable, rather than w, the
reader should recall the arguments made in Chapter 4 to justify the study of
a scalar ODE. Our first task is to find numerical approximations that can be
used to carry out the time integration of Eq. 6.3 to some given accuracy, where
accuracy can be measured either in a local or a global sense. We then face a
further task concerning the numerical stability of the resulting methods, but
we postpone such considerations to the next chapter.
In Chapter 2, we introduced the convention that the n subscript, or the
(n) superscript, always points to a discrete time value, and h represents the
time interval ∆t. Combining this notation with Eq. 6.3 gives
u0n = Fn = F (un , tn ) ,
tn = nh
Often we need a more sophisticated notation for intermediate time steps
involving temporary calculations denoted by ũ, ū, etc. For these we use the
notation
ũ0n+α = F̃n+α = F (ũn+α , tn + αh)
The choice of u0 or F to express the derivative in a scheme is arbitrary. They
are both commonly used in the literature on ODE’s.
The methods we study are to be applied to linear or nonlinear ODE’s, but
the methods themselves are formed by linear combinations of the dependent
variable and its derivative at various time intervals. They are represented
conceptually by
¢
¡
(6.4)
un+1 = f β1 u0n+1 , β0 u0n , β−1 u0n−1 , · · · , α0 un , α−1 un−1 , · · ·
With an appropriate choice of the α’s and β’s, these methods can be constructed to give a local Taylor series accuracy of any order. The methods are
said to be explicit if β1 = 0 and implicit otherwise. An explicit method is
one in which the new predicted solution is only a function of known data,
for example, u0n , u0n−1 , un , and un−1 for a method using two previous time
levels, and therefore the time advance is simple. For an implicit method, the
new predicted solution is also a function of the time derivative at the new
time level, that is, u0n+1 . As we shall see, for systems of ODE’s and nonlinear
problems, implicit methods require more complicated strategies to solve for
un+1 than explicit methods.
6.2 Converting Time-Marching Methods to O∆E’s
83
6.2 Converting Time-Marching Methods to O∆E’s
Examples of some very common forms of methods used for time-marching
general ODE’s are:
un+1 = un + hu0n
(6.5)
un+1 = un + hu0n+1
(6.6)
and
ũn+1 = un + hu0n
1
un+1 = [un + ũn+1 + hũ0n+1 ]
2
(6.7)
According to the conditions presented under Eq. 6.4, the first and third of
these are examples of explicit methods. We refer to them as the explicit Euler
method and the MacCormack predictor-corrector method,1 respectively. The
second is implicit and referred to as the implicit (or backward) Euler method.
These methods are simple recipes for the time advance of a function in
terms of its value and the value of its derivative, at given time intervals. The
material presented in Chapter 4 develops a basis for evaluating such methods
by introducing the concept of the representative equation
du
= u0 = λu + aeµt
dt
(6.8)
written here in terms of the dependent variable, u. The value of this equation arises from the fact that, by applying a time-marching method, we can
analytically convert such a linear ODE into a linear O∆E . The latter are
subject to a whole body of analysis that is similar in many respects to, and
just as powerful as, the theory of ODE’s. We next consider examples of this
conversion process and then go into the general theory on solving O∆E’s.
Apply the simple explicit Euler scheme, Eq. 6.5, to Eq. 6.8. There results
un+1 = un + h(λun + aeµhn )
or
un+1 − (1 + λh)un = haeµhn
(6.9)
Eq. 6.9 is a linear O∆E , with constant coefficients, expressed in terms of the
dependent variable un and the independent variable n. As another example,
applying the implicit Euler method, Eq. 6.6, to Eq. 6.8, we find
1
Here we give only MacCormack’s time-marching method. The method commonly
referred to as MacCormack’s method, which is a fully-discrete method, will be
presented in Section 11.3
84
6. TIME-MARCHING METHODS FOR ODE’S
³
´
un+1 = un + h λun+1 + aeµh(n+1)
or
(1 − λh)un+1 − un = heµh · aeµhn
(6.10)
As a final example, the predictor-corrector sequence, Eq. 6.7, gives
ũn+1 − (1 + λh)un
= aheµhn
1
1
1
− (1 + λh)ũn+1 + un+1 − un = aheµh(n+1)
2
2
2
(6.11)
which is a coupled set of linear O∆E’s with constant coefficients. Note that
the first line in Eq. 6.11 is identical to Eq. 6.9, since the predictor step in
Eq. 6.7 is simply the explicit Euler method. The second line in Eq. 6.11 is
obtained by noting that
ũ0n+1 = F (ũn+1 , tn + h)
= λũn+1 + aeµh(n+1)
(6.12)
Now we need to develop techniques for analyzing these difference equations so that we can compare the merits of the time-marching methods that
generated them.
6.3 Solution of Linear O∆E’s With Constant
Coefficients
The techniques for solving linear difference equations with constant coefficients is as well developed as that for ODE’s, and the theory follows a
remarkably parallel path. This is demonstrated by repeating some of the
developments in Section 4.2, but for difference rather than differential equations.
6.3.1 First- and Second-Order Difference Equations
First-Order Equations. The simplest nonhomogeneous O∆E of interest is
given by the single first-order equation
un+1 = σun + abn
(6.13)
where σ, a, and b are, in general, complex parameters. The independent
variable is now n rather than t, and since the equations are linear and have
constant coefficients, σ is not a function of either n or u. The exact solution
of Eq. 6.13 is
6.3 Solution of Linear O∆E’s With Constant Coefficients
85
n
ab
b−σ
where c1 is a constant determined by the initial conditions. In terms of the
initial value of u it can be written
bn − σ n
un = u0 σ n + a
b−σ
un = c1 σ n +
Just as in the development of Eq. 4.10, one can readily show that the solution
of the defective case, (b = σ),
un+1 = σun + aσ n
is
¤
£
un = u0 + anσ −1 σ n
This can all be easily verified by substitution.
Second-Order Equations. The homogeneous form of a second-order difference equation is given by
un+2 + a1 un+1 + a0 un = 0
(6.14)
d
dt
Instead of the differential operator D ≡
used for ODE’s, we use for O∆E’s
the difference operator E (commonly referred to as the displacement or shift
operator) and defined formally by the relations
un+1 = Eun
,
un+k = E k un
Further notice that the displacement operator also applies to exponents, thus
bα · bn = bn+α = E α · bn
where α can be any fraction or irrational number.
The roles of D and E are the same insofar as once they have been introduced to the basic equations, the value of u(t) or un can be factored out.
Thus Eq. 6.14 can now be re-expressed in an operational notation as
(E 2 + a1 E + a0 )un = 0
(6.15)
which must be zero for all un . Eq. 6.15 is known as the operational form of Eq.
6.14. The operational form contains a characteristic polynomial P (E) which
plays the same role for difference equations that P (D) played for differential
equations; that is, its roots determine the solution to the O∆E. In the analysis
of O∆E’s, we label these roots σ1 , σ2 , · · ·, etc, and refer to them as the σroots. They are found by solving the equation P (σ) = 0. In the simple example
given above, there are just two σ roots, and in terms of them the solution
can be written
un = c1 (σ1 )n + c2 (σ2 )n
(6.16)
where c1 and c2 depend upon the initial conditions. The fact that Eq. 6.16 is
a solution to Eq. 6.14 for all c1 , c2 , and n should be verified by substitution.
86
6. TIME-MARCHING METHODS FOR ODE’S
6.3.2 Special Cases of Coupled First-Order Equations
A Complete System. Coupled, first-order, linear, homogeneous difference
equations have the form
(n+1)
= c11 u1 + c12 u2
(n+1)
= c21 u1 + c22 u2
u1
u2
(n)
(n)
(n)
(n)
(6.17)
which can also be written
~un+1 = C ~un
,
h
iT
~un = u(n) , u(n)
1
2
·
,
C=
c11
c21
c12
c22
¸
The operational form of Eq. 6.17 can be written
·
(c11 − E)
c12
c21
(c22 − E)
¸·
u1
u2
¸(n)
= [ C − E I ]~un = 0
which must be zero for all u1 and u2 . Again we are led to a characteristic
polynomial, this time having the form P (E) = det [ C − E I ]. The σ-roots
are found from
·
¸
(c11 − σ)
c12
P (σ) = det
=0
c21
(c22 − σ)
Obviously, the σk are the eigenvalues of C, and following the logic of
Section 4.2, if ~x are its eigenvectors, the solution of Eq. 6.17 is
~un =
2
X
n
ck (σk ) ~xk
k=1
where ck are constants determined by the initial conditions.
A Defective System. The solution of O∆E’s with defective eigensystems
follows closely the logic in Section 4.2.2 for defective ODE’s. For example,
one can show that the solution to

 
 
ūn+1
σ
ūn
 ûn+1  =  1 σ
  ûn 
un+1
1 σ
un
is
ūn = ū0 σ n
£
¤
ûn = û0 + ū0 nσ −1 σ n
¸
·
n(n − 1) −2 n
σ
σ
un = u0 + û0 nσ −1 + ū0
2
(6.18)
6.4 Solution of the Representative O∆E’s
87
6.4 Solution of the Representative O∆E’s
6.4.1 The Operational Form and its Solution
Examples of the nonhomogeneous, linear, first-order ordinary difference equations produced by applying a time-marching method to the representative
equation are given by Eqs. 6.9 to 6.11. Using the displacement operator, E,
these equations can be written
"
[E − (1 + λh)]un = h · aeµhn
(6.19)
[(1 − λh)E − 1]un = h · E · aeµhn
(6.20)
E
−(1 + λh)
− 12 (1 + λh)E E − 12
#· ¸
"
#
1
ũ
· aeµhn
=h· 1
u n
E
2
(6.21)
All three of these equations are subsets of the operational form of the
representative O∆E
P (E)un = Q(E) · aeµhn
(6.22)
which is produced by applying time-marching methods to the representative
ODE, Eq. 4.45. We can express in terms of Eq. 6.22 all manner of standard time-marching methods having multiple time steps and various types
of intermediate predictor-corrector families. The terms P (E) and Q(E) are
polynomials in E referred to as the characteristic polynomial and the particular polynomial, respectively.
The general solution of Eq. 6.22 can be expressed as
un =
K
X
n
ck (σk ) + aeµhn ·
k=1
Q(eµh )
P (eµh )
(6.23)
where σk are the K roots of the characteristic polynomial, P (σ) = 0. When
determinants are involved in the construction of P (E) and Q(E), as would
be the case for Eq. 6.21, the ratio Q(E)/P (E) can be found by Cramer’s rule.
Keep in mind that for methods such as in Eq. 6.21 there are multiple (two
in this case) solutions, one for un and one for ũn , and we are usually only
interested in the final solution un . Notice also, the important subset of this
solution which occurs when µ = 0, representing a time-invariant particular
solution, or a steady state. In such a case
un =
K
X
k=1
n
ck (σk ) + a ·
Q(1)
P (1)
88
6. TIME-MARCHING METHODS FOR ODE’S
6.4.2 Examples of Solutions to Time-Marching O∆E’s
As examples of the use of Eqs. 6.22 and 6.23, we derive the solutions of Eqs.
6.19 to 6.21. For the explicit Euler method, Eq. 6.19, we have
P (E) = E − 1 − λh
Q(E) = h
(6.24)
and the solution of its representative O∆E follows immediately from Eq. 6.23:
un = c1 (1 + λh)n + aeµhn ·
e
µh
h
− 1 − λh
For the implicit Euler method, Eq. 6.20, we have
P (E) = (1 − λh)E − 1
Q(E) = hE
so
µ
un = c1
1
1 − λh
¶n
+ aeµhn ·
(6.25)
heµh
(1 − λh)eµh − 1
In the case of the coupled predictor-corrector equations, Eq. 6.21, one solves
for the final family un (one can also find a solution for the intermediate family
ũ), and there results
·
¸
µ
¶
E
−(1 + λh)
1 2 2
P (E) = det
=
E
E
−
1
−
λh
−
λ
h
− 21 (1 + λh)E
E − 12
2
·
¸
1
E
h
Q(E) = det
= hE(E + 1 + λh)
− 21 (1 + λh)E 12 hE
2
The σ-root is found from
µ
¶
1
P (σ) = σ σ − 1 − λh − λ2 h2 = 0
2
which has only one nontrivial root (σ = 0 is simply a shift in the reference
index). The complete solution can therefore be written
µ
¶n
1 2 2
un = c1 1 + λh + λ h
+ aeµhn ·
2
¢
1 ¡ µh
h e + 1 + λh
2
1
µh
e − 1 − λh − λ2 h2
2
(6.26)
6.5 The λ − σ Relation
89
6.5 The λ − σ Relation
6.5.1 Establishing the Relation
We have now been introduced to two basic kinds of roots, the λ-roots and
the σ-roots. The former are the eigenvalues of the A matrix in the ODE’s
found by space differencing the original PDE, and the latter are the roots of
the characteristic polynomial in a representative O∆E found by applying a
time-marching method to the representative ODE. There is a fundamental
relation between the two which can be used to identify many of the essential
properties of a time-march method. This relation is first demonstrated by
developing it for the explicit Euler method.
First we make use of the semi-discrete approach to find a system of ODE’s
and then express its solution in the form of Eq. 4.27. Remembering that
t = nh, one can write
¡
¢n
¡
¢n
¡
¢n
~u(t) = c1 eλ1 h ~x1 + · · · + cm eλm h ~xm + · · · + cM eλM h ~xM + P.S.
(6.27)
where for the present we are not interested in the form of the particular
solution (P.S.). Now the explicit Euler method produces for each λ-root, one
σ-root, which is given by σ = 1 + λh. So if we use the Euler method for the
time advance of the ODE’s, the solution2 of the resulting O∆E is
n
n
n
~un = c1 (σ1 ) ~x1 + · · · + cm (σm ) ~xm + · · · + cM (σM ) ~xM + P.S.
(6.28)
where the cm and the ~xm in the two equations are identical and σm =
(1 + λm h). Comparing Eq. 6.27 and Eq. 6.28, we see a correspondence between σm and eλm h . Since the value of eλh can be expressed in terms of the
series
1
1
1
eλh = 1 + λh + λ2 h2 + λ3 h3 + · · · + λn hn + · · ·
2
6
n!
the truncated expansion σ = 1 + λh is a reasonable3 approximation for small
enough λh.
Suppose, instead of the Euler method, we use the leapfrog method for the
time advance, which is defined by
un+1 = un−1 + 2hu0n
(6.29)
Applying Eq. 6.29 to Eq. 6.8, we have the characteristic polynomial P (E) =
E 2 − 2λhE − 1, so that for every λ the σ must satisfy the relation
2
σm
− 2λm hσm − 1 = 0
2
3
Based on Section 4.4.
The error is O(λ2 h2 ).
(6.30)
90
6. TIME-MARCHING METHODS FOR ODE’S
Now we notice that each λ produces two σ-roots. For one of these we find
p
σm = λm h + 1 + λ2m h2
(6.31)
1 2 2 1 4 4
= 1 + λm h + λm h − λm h + · · ·
(6.32)
2
8
This is p
an approximation to eλm h with an error O(λ3 h3 ). The other root,
2 h2 , will be discussed in Section 6.5.3.
λm h − 1 + λm
6.5.2 The Principal σ-Root
Based on the above we make the following observation:
Application of the same time-marching method to all of
the equations in a coupled system of linear ODE’s in the
form of Eq. 4.6 always produces one σ-root for every λ-root
that satisfies the relation
(6.33)
¡
¢
1
1
σ = 1 + λh + λ2 h2 + · · · + λk hk + O hk+1
2
k!
where k is the order of the time-marching method.
We refer to the root that has the above property as the principal σ-root,
and designate it (σm )1 . The above property can be stated regardless of the
details
of
¡
¢ the time-marching method, knowing only that its leading error
¡ ¢ is
O hk+1 . Thus the principal root is an approximation to eλh up to O hk .
Note that a second-order approximation to a derivative written in the form
(δt u)n =
1
(un+1 − un−1 )
2h
(6.34)
has a leading truncation error which is O(h2 ), while the second-order timemarching method which results from this approximation, which is the leapfrog
method:
un+1 = un−1 + 2hu0n
(6.35)
has a leading truncation error O(h3 ). This arises simply because of our notation for the time-marching method in which we have multiplied through by
h to get an approximation for the function un+1 rather than the derivative
as in Eq. 6.34. The following example makes this clear. Consider a solution
obtained at a given time T using a second-order time-marching method with
a time step h. Now consider the solution obtained using the same method
with a time step h/2. Since the error per time step is O(h3 ), this is reduced
by a factor of eight (considering the leading term only). However, twice as
many time steps are required to reach the time T . Therefore the error at
the end of the simulation is reduced by a factor of four, consistent with a
second-order approximation.
6.5 The λ − σ Relation
91
6.5.3 Spurious σ-Roots
We saw from Eq. 6.30 that the λ−σ relation for the leapfrog method produces
two σ-roots for each λ. One of these we identified as the principal root, which
always has the property given in 6.33. The other is referred to as a spurious
σ-root and designated (σm )2 . In general, the λ − σ relation produced by a
time-marching scheme can result in multiple σ-roots all of which, except for
the principal one, are spurious. All spurious roots are designated (σm )k where
k = 2, 3, · · ·. No matter whether a σ-root is principal or spurious, it is always
some algebraic function of the product λh. To express this fact we use the
notation σ = σ(λh).
If a time-marching method produces spurious σ-roots, the solution for the
O∆E in the form shown in Eq. 6.28 must be modified. Following again the
message of Section 4.4, we have
n
n
n
~un = c11 (σ1 )1 ~x1 + · · · + cm1 (σm )1 ~xm + · · · + cM 1 (σM )1 ~xM + P.S.
n
n
n
+c12 (σ1 ) ~x1 + · · · + cm2 (σm ) ~xm + · · · + cM 2 (σM ) ~xM
2
2
2
n
n
n
+c13 (σ1 )3 ~x1 + · · · + cm3 (σm )3 ~xm + · · · + cM 3 (σM )3 ~xM
+etc., if there are more spurious roots
(6.36)
Spurious roots arise if a method uses data from time level n−1 or earlier to
advance the solution from time level n to n + 1. Such roots originate entirely
from the numerical approximation of the time-marching method and have
nothing to do with the ODE being solved. However, generation of spurious
roots does not, in itself, make a method inferior. In fact, many very accurate
methods in practical use for integrating some forms of ODE’s have spurious
roots.
It should be mentioned that methods with spurious roots are not selfstarting. For example, if there is one spurious root to a method, all of the
coefficients (cm )2 in Eq. 6.36 must be initialized by some starting procedure.
The initial vector ~u0 does not provide enough data to initialize all of the
coefficients. This results because methods which produce spurious roots require data from time level n − 1 or earlier. For example, the leapfrog method
requires ~un−1 and thus cannot be started using only ~un .
Presumably (i.e., if one starts the method properly4 ) the spurious coefficients are all initialized with very small magnitudes, and presumably the
magnitudes of the spurious roots themselves are all less than one (see Chapter 7). Then the presence of spurious roots does not contaminate the answer.
That is, after some finite time, the amplitude of the error associated with
the spurious roots is even smaller then when it was initialized. Thus while
spurious roots must be considered in stability analysis, they play virtually no
role in accuracy analysis.
4
Normally using a self-starting method for the first step or steps, as required.
92
6. TIME-MARCHING METHODS FOR ODE’S
6.5.4 One-Root Time-Marching Methods
There are a number of time-marching methods that produce only one σ-root
for each λ-root. We refer to them as one-root methods. They are also called
one-step methods. They have the significant advantage of being self-starting
which carries with it the very useful property that the time-step interval can
be changed at will throughout the marching process. Three one-root methods
were analyzed in Section 6.4.2. A popular method having this property, the
so-called θ-method, is given by the formula
£
¤
un+1 = un + h (1 − θ)u0n + θu0n+1
The θ-method represents the explicit Euler (θ = 0), the trapezoidal (θ = 12 ),
and the implicit Euler methods (θ = 1), respectively. Its λ − σ relation is
σ=
1 + (1 − θ)λh
1 − θλh
It is instructive to compare the exact solution to a set of ODE’s (with
a complete eigensystem) having time-invariant forcing terms with the exact
solution to the O∆E’s for one-root methods. These are
¡
¢
¡
¢
¡
¢
~u(t) = c1 eλ1 h n ~x1 + · · · + cm eλm h n ~xm + · · · + cM eλM h n ~xM + A−1~f
~un = c1 (σ1 )n ~x1 + · · · + cm (σm )n ~xm + · · · + cM (σM )n ~xM + A−1~f (6.37)
respectively. Notice that when t and n = 0, these equations are identical, so
that all the constants, vectors, and matrices are identical except the ~u and
the terms inside the parentheses on the right hand sides. The only error made
by introducing the time marching is the error that σ makes in approximating
eλh .
6.6 Accuracy Measures of Time-Marching Methods
6.6.1 Local and Global Error Measures
There are two broad categories of errors that can be used to derive and
evaluate time-marching methods. One is the error made in each time step.
This is a local error such as that found from a Taylor table analysis (see
Section 3.4). It is usually used as the basis for establishing the order of a
method. The other is the error determined at the end of a given event which
has covered a specific interval of time composed of many time steps. This is
a global error. It is useful for comparing methods, as we shall see in Chapter
8.
It is quite common to judge a time-marching method on the basis of results
found from a Taylor table. However, a Taylor series analysis is a very limited
tool for finding the more subtle properties of a numerical time-marching
method. For example, it is of no use in:
6.6 Accuracy Measures of Time-Marching Methods
93
• finding spurious roots.
• evaluating numerical stability and separating the errors in phase and amplitude.
• analyzing the particular solution of predictor-corrector combinations.
• finding the global error.
The latter three of these are of concern to us here, and to study them we
make use of the material developed in the previous sections of this chapter.
Our error measures are based on the difference between the exact solution to
the representative ODE, given by
u(t) = ceλt +
aeµt
µ−λ
(6.38)
and the solution to the representative O∆E’s, including only the contribution
from the principal root, which can be written as
n
un = c1 (σ1 ) + aeµhn ·
Q(eµh )
P (eµh )
(6.39)
6.6.2 Local Accuracy of the Transient Solution (erλ , |σ| , erω )
Transient error. The particular choice of an error measure, either local or
global, is to some extent arbitrary. However, a necessary condition for the
choice should be that the measure can be used consistently for all methods.
In the discussion of the λ-σ relation we saw that all time-marching methods
produce a principal σ-root for every λ-root that exists in a set of linear ODE’s.
Therefore, a very natural local error measure for the transient solution is
the value of the difference between solutions based on these two roots. We
designate this by erλ and make the following definition
erλ ≡ eλh − σ1
The leading error term can be found by expanding in a Taylor series and
choosing the first nonvanishing term. This is similar to the error found from
a Taylor table. The order of the method is the last power of λh matched
exactly.
Amplitude and Phase Error. Suppose a λ eigenvalue is imaginary. Such
can indeed be the case when we study the equations governing periodic convection, which produces harmonic motion. For such cases it is more meaningful to express the error in terms of amplitude and phase. Let λ = iω where ω
is a real number representing a frequency. Then the numerical method must
produce a principal σ-root that is complex and expressible in the form
σ1 = σr + iσi ≈ eiωh
(6.40)
94
6. TIME-MARCHING METHODS FOR ODE’S
From this it follows that the local error in amplitude is measured by the
deviation of |σ1 | from unity, that is
q
era = 1 − |σ1 | = 1 −
(σ1 )2r + (σ1 )2i
and the local error in phase can be defined as
erω ≡ ωh − tan−1 [(σ1 )i /(σ1 )r )]
(6.41)
Amplitude and phase errors are important measures of the suitability of timemarching methods for convection and wave propagation phenomena.
The approach to error analysis described in Section 3.5 can be extended
to the combination of a spatial discretization and a time-marching method
applied to the linear convection equation. The principal root, σ1 (λh), is
found using λ = −iaκ∗ , where κ∗ is the modified wavenumber of the spatial discretization. Introducing the Courant number, Cn = ah/∆x, we have
λh = −iCn κ∗ ∆x. Thus one can obtain values of the principal root over the
range 0 ≤ κ∆x ≤ π for a given value of the Courant number. The above
expression for erω can be normalized to give the error in the phase speed, as
follows
erp =
erω
tan−1 [(σ1 )i /(σ1 )r )]
=1+
ωh
Cn κ∆x
(6.42)
where ω = −aκ. A positive value of erp corresponds to phase lag (the numerical phase speed is too small), while a negative value corresponds to phase
lead (the numerical phase speed is too large).
6.6.3 Local Accuracy of the Particular Solution (erµ )
The numerical error in the particular solution is found by comparing the
particular solution of the ODE with that for the O∆E. We have found these
to be given by
1
P.S.(ODE) = aeµt ·
(µ − λ)
and
P.S.(O∆E) = aeµt ·
Q(eµh )
P (eµh )
respectively. For a measure of the local error in the particular solution we
introduce the definition
½
¾
P.S.(O∆E)
erµ ≡ h
−1
(6.43)
P.S.(ODE)
The multiplication by h converts the error from a global measure to a local
one, so that the order of erλ and erµ are consistent. In order to determine
6.6 Accuracy Measures of Time-Marching Methods
95
the leading error term, Eq. 6.43 can be written in terms of the characteristic
and particular polynomials as
erµ =
©
¡ ¢
¡ ¢ª
co
· (µ − λ)Q eµh − P eµh
µ−λ
(6.44)
and expanded in a Taylor series, where
h(µ − λ)
¡ ¢
h→0 P eµh
co = lim
The value of co is a method-dependent constant that is often equal to one.
If the forcing function is independent of time, µ is equal to zero, and for this
case, many numerical methods generate an erµ that is also zero.
The algebra involved in finding the order of erµ can be quite tedious.
However, this order is quite important in determining the true order of a
time-marching method by the process that has been outlined. An illustration
of this is given in the section on Runge-Kutta methods.
6.6.4 Time Accuracy For Nonlinear Applications
In practice, time-marching methods are usually applied to nonlinear ODE’s,
and it is necessary that the advertised order of accuracy be valid for the
nonlinear cases as well as for the linear ones. A necessary condition for this
to occur is that the local accuracies of both the transient and the particular
solutions be of the same order. More precisely, a time-marching method is
said to be of order k if
erλ
= c1 · (λh)k1 +1
erµ
= c2 · (λh)k2 +1
where k = smallest of(k1 , k2 )
(6.45)
(6.46)
(6.47)
The reader should be aware that this is not sufficient. For example, to derive
all of the necessary conditions for the fourth-order Runge-Kutta method presented later in this chapter the derivation must be performed for a nonlinear
ODE. However, the analysis based on a linear nonhomogeneous ODE produces the appropriate conditions for the majority of time-marching methods
used in CFD.
6.6.5 Global Accuracy
In contrast to the local error measures which have just been discussed, we
can also define global error measures. These are useful when we come to the
evaluation of time-marching methods for specific purposes. This subject is
covered in Chapter 8 after our introduction to stability in Chapter 7.
96
6. TIME-MARCHING METHODS FOR ODE’S
Suppose we wish to compute some time-accurate phenomenon over a fixed
interval of time using a constant time step. We refer to such a computation
as an “event”. Let T be the fixed time of the event and h be the chosen step
size. Then the required number of time steps, is N , given by the relation
T = Nh
Global error in the transient. A natural extension of erλ to cover the
error in an entire event is given by
Erλ ≡ eλT − (σ1 (λh))
N
(6.48)
Global error in amplitude and phase. If the event is periodic, we are
more concerned with the global error in amplitude and phase. These are given
by
Era = 1 −
µq
¶N
(σ1 )2r + (σ1 )2i
(6.49)
and
·
µ
¶¸
(σ1 )i
Erω ≡ N ωh − tan−1
(σ1 )r
= ωT − N tan−1 [(σ1 )i /(σ1 )r ]
(6.50)
Global error in the particular solution. Finally, the global error in the
particular solution follows naturally by comparing the solutions to the ODE
and the O∆E. It can be measured by
¡ ¢
Q eµh
Erµ ≡ (µ − λ) ¡ µh ¢ − 1
P e
6.7 Linear Multistep Methods
In the previous sections, we have developed the framework of error analysis for time advance methods and have randomly introduced a few methods
without addressing motivational, developmental, or design issues. In the subsequent sections, we introduce classes of methods along with their associated
error analysis. We shall not spend much time on development or design of
these methods, since most of them have historic origins from a wide variety
of disciplines and applications. The Linear Multistep Methods (LMM’s) are
probably the most natural extension to time marching of the space differencing schemes introduced in Chapter 3 and can be analyzed for accuracy or
designed using the Taylor table approach of Section 3.4.
6.7 Linear Multistep Methods
97
6.7.1 The General Formulation
When applied to the nonlinear ODE
du
= u0 = F (u, t)
dt
all linear multistep methods can be expressed in the general form
1
X
1
X
αk un+k = h
k=1−K
βk Fn+k
(6.51)
k=1−K
where the notation for F is defined in Section 6.1. The methods are said to
be linear because the α’s and β’s are independent of u and n, and they are
said to be K-step because K time-levels of data are required to march the
solution one time-step, h. They are explicit if β1 = 0 and implicit otherwise.
When Eq. 6.51 is applied to the representative equation, Eq. 6.8, and the
result is expressed in operational form, one finds
à 1
!
à 1
!
X
X
k
k
αk E un = h
βk E (λun + aeµhn )
(6.52)
k=1−K
k=1−K
We recall from Section 6.5.2 that a time-marching method when applied to
the representative equation must provide a σ-root, labeled σ1 , that approximates eλh through the order of the method. The condition referred to as
consistency simply means that σ → 1 as h → 0, and it is certainly a necessary condition for the accuracy of any time-marching method. We can also
agree that, to be of any value in time accuracy, a method should at least be
first-order accurate, that is σ → (1 + λh) as h → 0. One can show that these
conditions are met by any method represented by Eq. 6.51 if
X
X
X
αk = 0 and
βk =
(K + k − 1)αk
k
k
k
Since both sides of Eq. 6.51 can be multiplied by an arbitrary constant, these
methods are often “normalized” by requiring
X
βk = 1
k
Under this condition, co = 1 in Eq. 6.44.
6.7.2 Examples
There are many special explicit and implicit forms of linear multistep methods. Two well-known families of them, referred to as Adams-Bashforth (explicit) and Adams-Moulton (implicit), can be designed using the Taylor table
98
6. TIME-MARCHING METHODS FOR ODE’S
approach of Section 3.4. The Adams-Moulton family is obtained from Eq.
6.51 with
α1 = 1,
α0 = −1,
αk = 0,
k = −1, −2, · · ·
(6.53)
The Adams-Bashforth family has the same α’s with the additional constraint
that β1 = 0. The three-step Adams-Moulton method can be written in the
following form
un+1 = un + h(β1 u0n+1 + β0 u0n + β−1 u0n−1 + β−2 u0n−2 )
A Taylor table for Eq. 6.54 can be generated as shown in Table 6.1.
This leads to the linear system


  
1 1 1
1
β1
1
 2 0 −2 −4   β0   1 


= 
 3 0 3 12   β−1   1 
4 0 −4 −32
β−2
1
(6.54)
(6.55)
to solve for the β’s, resulting in
β1 = 9/24,
β0 = 19/24,
β−1 = −5/24,
β−2 = 1/24
(6.56)
which produces a method which is fourth-order accurate.5 With β1 = 0 one
obtains


  
1 1 1
β0
1
 0 −2 −4   β−1  =  1 
(6.57)
0 3 12
β−2
1
giving
β0 = 23/12,
β−1 = −16/12,
β−2 = 5/12
(6.58)
This is the third-order Adams-Bashforth method.
A list of simple methods, some of which are very common in CFD applications, is given below together with identifying names that are sometimes
associated with them. In the following material AB(n) and AM(n) are used as
abbreviations for the (n)th order Adams-Bashforth and (n)th order AdamsMoulton methods. One can verify that the Adams type schemes given below
satisfy Eqs. 6.55 and 6.57 up to the order of the method.
Explicit Methods.
un+1
un+1
un+1
un+1
5
= un + hu0n
Euler
= un−1 + 2hu0n
Leapfrog
£
¤
= un + 12 h 3u0n − u0n−1
AB2
£
¤
h
= un + 12 23u0n − 16u0n−1 + 5u0n−2
AB3
Recall from Section 6.5.2 that a kth-order time-marching method has a leading
truncation error term which is O(hk+1 ).
− hβ−2 u0n−2
−1
−(−2)1 β−2
−(−2)0 β−2
−(−2)3 β−2 16
β−1 16
−β−1 21
β−1
−(−2)2 β−2 21
−β1 16
h4 · u0000
n
1
24
−β1 12
h3 · u000
n
1
6
−β1
h2 · u00n
1
2
−β1
−β0
−β−1
h · u0n
1
Table 6.1. Taylor table for the Adams-Moulton three-step linear multistep method.
− un
− hβ1 u0n+1
− hβ0 u0n
− hβ−1 u0n−1
un+1
un
1
6.7 Linear Multistep Methods
99
100
6. TIME-MARCHING METHODS FOR ODE’S
Implicit Methods.
un+1
un+1
un+1
un+1
= un + hu0n+1
Implicit Euler
£ 0
¤
1
0
= un + 2 h un + un+1
Trapezoidal (AM2)
£
¤
1
0
= 3 4un − un−1 + 2hun+1
2nd-order Backward
£ 0
¤
h
0
0
= un + 12 5un+1 + 8un − un−1
AM3
6.7.3 Two-Step Linear Multistep Methods
High-resolution CFD problems usually require very large data sets to store
the spatial information from which the time derivative is calculated. This
limits the interest in multistep methods to about two time levels. The most
general two-step linear multistep method (i.e., K = 2 in Eq. 6.51) that is at
least first-order accurate can be written as
¤
(1 + ξ)un+1 = [(1 + 2ξ)un − ξun−1 ] + h [ θu0n+1 + (1 − θ + ϕ)u0n − ϕu0n−1
(6.59)
Clearly the methods are explicit if θ = 0 and implicit otherwise. A list of
methods contained in Eq. 6.59 is given in Table 6.2. Notice that the Adams
methods have ξ = 0, which corresponds to α−1 = 0 in Eq. 6.51. Methods
with ξ = −1/2, which corresponds to α0 = 0 in Eq. 6.51, are known as Milne
methods.
θ
ξ
ϕ
0
0
0
1
0
0
1/2
0
0
1
1/2
0
3/4
0
−1/4
1/3 −1/2 −1/3
1/2 −1/2 −1/2
5/9 −1/6 −2/9
0 −1/2
0
0
0
1/2
0 −5/6 −1/3
1/3 −1/6
0
5/12
0
1/12
1/6 −1/2 −1/6
Method
Order
Euler
Implicit Euler
Trapezoidal or AM2
2nd Order Backward
Adams type
Lees
Two–step trapezoidal
A–contractive
Leapfrog
AB2
Most accurate explicit
Third–order implicit
AM3
Milne
1
1
2
2
2
2
2
2
2
2
3
3
3
4
Table 6.2. Some linear one- and two-step methods; see Eq. 6.59.
6.8 Predictor-Corrector Methods
101
One can show after a little algebra that both erµ and erλ are reduced to
0(h3 ) (i.e., the methods are 2nd-order accurate) if
ϕ=ξ−θ+
1
2
The class of all 3rd-order methods is determined by imposing the additional
constraint
5
ξ = 2θ −
6
Finally a unique fourth-order method is found by setting θ = −ϕ = −ξ/3 =
1.
6
6.8 Predictor-Corrector Methods
There are a wide variety of predictor-corrector schemes created and used for a
variety of purposes. Their use in solving ODE’s is relatively easy to illustrate
and understand. Their use in solving PDE’s can be much more subtle and
demands concepts6 which have no counterpart in the analysis of ODE’s.
Predictor-corrector methods constructed to time-march linear or nonlinear
ODE’s are composed of sequences of linear multistep methods, each of which
is referred to as a family in the solution process. There may be many families
in the sequence, and usually the final family has a higher Taylor-series order
of accuracy than the intermediate ones. Their use is motivated by ease of
application and increased efficiency, where measures of efficiency are discussed
in the next two chapters.
A simple one-predictor, one-corrector example is given by
ũn+α = un + αhu0n
¤
£
un+1 = un + h β ũ0n+α + γu0n
(6.60)
where the parameters α, β and γ are arbitrary parameters to be determined.
One can analyze this sequence by applying it to the representative equation
and using the operational techniques outlined in Section 6.4. It is easy to
show, following the example leading to Eq. 6.26, that
£
¤
P (E) = E α · E − 1 − (γ + β)λh − αβλ2 h2
(6.61)
α
α
Q(E) = E · h · [βE + γ + αβλh]
(6.62)
Considering only local accuracy, one is led, by following the discussion in
Section 6.6, to the following observations. For the method to be second-order
accurate both erλ and erµ must be O(h3 ). For this to hold for erλ , it is obvious
from Eq. 6.61 that
6
Such as alternating direction, fractional-step, and hybrid methods.
102
6. TIME-MARCHING METHODS FOR ODE’S
1
2
which provides two equations for three unknowns. The situation for erµ requires some algebra, but it is not difficult to show using Eq. 6.44 that the same
conditions also make it O(h3 ). One concludes, therefore, that the predictorcorrector sequence
γ+β =1 ;
αβ =
ũn+α = un + αhu0n
·µ ¶
¶ ¸
µ
1
1 0
2α − 1 0
un+1 = un + h
ũ
un
+
2
α n+α
α
(6.63)
is a second-order accurate method for any α.
A classical predictor-corrector sequence is formed by following an AdamsBashforth predictor of any order with an Adams-Moulton corrector having an
order one higher. The order of the combination is then equal to the order of
the corrector. If the order of the corrector is (k), we refer to these as ABM(k)
methods. The Adams-Bashforth-Moulton sequence for k = 3 is
¤
1 £
ũn+1 = un + h 3u0n − u0n−1
2
¤
h£ 0
5ũn+1 + 8u0n − u0n−1
un+1 = un +
12
(6.64)
Some simple, specific, second-order accurate methods are given below. The
Gazdag method, which we discuss in Chapter 8, is
¤
1 £
ũn+1 = un + h 3ũ0n − ũ0n−1
2
¤
1 £ 0
un+1 = un + h ũn + ũ0n+1
2
(6.65)
The Burstein method, obtained from Eq. 6.63 with α = 1/2 is
1
ũn+1/2 = un + hu0n
2
un+1 = un + hũ0n+1/2
(6.66)
and, finally, MacCormack’s method, presented earlier in this chapter, is
ũn+1 = un + hu0n
1
un+1 = [un + ũn+1 + hũ0n+1 ]
2
(6.67)
Note that MacCormack’s method can also be written as
ũn+1 = un + hu0n
1
un+1 = un + h[u0n + ũ0n+1 ]
2
from which it is clear that it is obtained from Eq. 6.63 with α = 1.
(6.68)
6.9 Runge-Kutta Methods
103
6.9 Runge-Kutta Methods
There is a special subset of predictor-corrector methods, referred to as RungeKutta methods,7 that produce just one σ-root for each λ-root such that σ(λh)
corresponds to the Taylor series expansion of eλh out through the order of
the method and then truncates. Thus for a Runge-Kutta method of order k
(up to 4th order), the principal (and only) σ-root is given by
1
1
σ = 1 + λh + λ2 h2 + · · · + λk hk
2
k!
(6.69)
It is not particularly difficult to build this property into a method, but, as
we pointed out in Section 6.6.4, it is not sufficient to guarantee kth order
accuracy for the solution of u0 = F (u, t) or for the representative equation.
To ensure kth order accuracy, the method must further satisfy the constraint
that
erµ = O(hk+1 )
(6.70)
and this is much more difficult.
The most widely publicized Runge-Kutta process is the one that leads to
the fourth-order method. We present it below in some detail. It is usually
introduced in the form
k1
k2
k3
k4
=
=
=
=
hF (un , tn )
hF (un + βk1 , tn + αh)
hF (un + β1 k1 + γ1 k2 , tn + α1 h)
hF (un + β2 k1 + γ2 k2 + δ2 k3 , tn + α2 h)
followed by
u(tn + h) − u(tn ) = µ1 k1 + µ2 k2 + µ3 k3 + µ4 k4
(6.71)
However, we prefer to present it using predictor-corrector notation. Thus, a
scheme entirely equivalent to 6.71 is
u
bn+α
ũn+α1
un+α2
un+1
= un + βhu0n
= un + β1 hu0n + γ1 hb
u0n+α
0
= un + β2 hun + γ2 hb
u0n+α + δ2 hũ0n+α1
= un + µ1 hu0n + µ2 hb
u0n+α + µ3 hũ0n+α1 + µ4 hu0n+α2
(6.72)
Appearing in Eqs. 6.71 and 6.72 are a total of 13 parameters which are to
be determined such that the method is fourth-order according to the requirements in Eqs. 6.69 and 6.70. First of all, the choices for the time samplings,
α, α1 , and α2 , are not arbitrary. They must satisfy the relations
7
Although implicit and multi-step Runge-Kutta methods exist, we will consider
only single-step, explicit Runge-Kutta methods here.
104
6. TIME-MARCHING METHODS FOR ODE’S
α =β
α1 = β1 + γ1
α2 = β2 + γ2 + δ2
(6.73)
The algebra involved in finding algebraic equations for the remaining 10 parameters is not trivial, but the equations follow directly from finding P (E)
and Q(E) and then satisfying the conditions in Eqs. 6.69 and 6.70. Using Eq.
6.73 to eliminate the β’s we find from Eq. 6.69 the four conditions
µ1 + µ2 + µ3 + µ4
= 1
µ2 α + µ3 α1 + µ4 α2 = 1/2
µ3 αγ1 + µ4 (αγ2 + α1 δ2 ) = 1/6
µ4 αγ1 δ2
= 1/24
(1)
(2)
(3)
(4)
(6.74)
These four relations guarantee that the five terms in σ exactly match the first
5 terms in the expansion of eλh . To satisfy the condition that erµ = O(h5 ),
we have to fulfill four more conditions
µ2 α2 + µ3 α12 + µ4 α22
µ2 α3 + µ3 α13 + µ4 α23
µ3 α2 γ1 + µ4 (α2 γ2 + α12 δ2 )
µ3 αα1 γ1 + µ4 α2 (αγ2 + α1 δ2 )
= 1/3
= 1/4
= 1/12
= 1/8
(3)
(4)
(4)
(4)
(6.75)
The number in parentheses at the end of each equation indicates the order
that is the basis for the equation. Thus if the first 3 equations in 6.74 and the
first equation in 6.75 are all satisfied, the resulting method would be thirdorder accurate. As discussed in Section 6.6.4, the fourth condition in Eq. 6.75
cannot be derived using the methodology presented here, which is based on a
linear nonhomogenous representative ODE. A more general derivation based
on a nonlinear ODE can be found in several books.8
There are eight equations in 6.74 and 6.75 which must be satisfied by the
10 unknowns. Since the equations are underdetermined, two parameters can
be set arbitrarily. Several choices for the parameters have been proposed, but
the most popular one is due to Runge. It results in the “standard” fourthorder Runge-Kutta method expressed in predictor-corrector form as
1
u
bn+1/2 = un + hu0n
2
1 0
ũn+1/2 = un + hb
u
2 n+1/2
un+1 = un + hũ0n+1/2
³
´
i
1 h
b0n+1/2 + ũ0n+1/2 + u0n+1
un+1 = un + h u0n + 2 u
6
8
(6.76)
The present approach based on a linear nonhomogeneous equation provides all
of the necessary conditions for Runge-Kutta methods of up to third order.
6.10 Implementation of Implicit Methods
105
Notice that this represents the simple sequence of conventional linear multistep methods referred to, respectively, as

Euler Predictor



Euler Corrector
≡ RK4
Leapfrog Predictor 


Milne Corrector
One can easily show that both the Burstein and the MacCormack methods
given by Eqs. 6.66 and 6.67 are second-order Runge-Kutta methods, and
third-order methods can be derived from Eqs. 6.72 by setting µ4 = 0 and
satisfying only the first three equations in 6.74 and the first equation in 6.75.
It is clear that for orders one through four, RK methods of order k require
k evaluations of the derivative function to advance the solution one time
step. We shall discuss the consequences of this in Chapter 8. Higher-order
Runge-Kutta methods can be developed, but they require more derivative
evaluations than their order. For example, a fifth-order method requires six
evaluations to advance the solution one step. In any event, storage requirements reduce the usefulness of Runge-Kutta methods of order higher than
four for CFD applications.
6.10 Implementation of Implicit Methods
We have presented a wide variety of time-marching methods and shown how
to derive their λ − σ relations. In the next chapter, we will see that these
methods can have widely different properties with respect to stability. This
leads to various trade-offs which must be considered in selecting a method
for a specific application. Our presentation of the time-marching methods in
the context of a linear scalar equation obscures some of the issues involved
in implementing an implicit method for systems of equations and nonlinear
equations. These are covered in this section.
6.10.1 Application to Systems of Equations
Consider first the numerical solution of our representative ODE
u0 = λu + aeµt
(6.77)
using the implicit Euler method. Following the steps outlined in Section 6.2,
we obtained
(1 − λh)un+1 − un = heµh · aeµhn
Solving for un+1 gives
(6.78)
106
6. TIME-MARCHING METHODS FOR ODE’S
un+1 =
1
(un + heµh · aeµhn )
1 − λh
(6.79)
This calculation does not seem particularly onerous in comparison with the
application of an explicit method to this ODE, requiring only an additional
division.
Now let us apply the implicit Euler method to our generic system of equations given by
~u0 = A~u − f~(t)
(6.80)
where ~u and f~ are vectors and we still assume that A is not a function of ~u
or t. Now the equivalent to Eq. 6.78 is
(I − hA)~un+1 − ~un = −hf~(t + h)
(6.81)
~un+1 = (I − hA)−1 [~un − hf~(t + h)]
(6.82)
or
The inverse is not actually performed, but rather we solve Eq. 6.81 as a
linear system of equations. For our one-dimensional examples, the system
of equations which must be solved is tridiagonal (e.g., for biconvection,
A = −aBp (−1, 0, 1)/2∆x), and hence its solution is inexpensive, but in multidimensions the bandwidth can be very large. In general, the cost per time
step of an implicit method is larger than that of an explicit method. The primary area of application of implicit methods is in the solution of stiff ODE’s,
as we shall see in Chapter 8.
6.10.2 Application to Nonlinear Equations
Now consider the general nonlinear scalar ODE given by
du
= F (u, t)
dt
(6.83)
Application of the implicit Euler method gives
un+1 = un + hF (un+1 , tn+1 )
(6.84)
This is a nonlinear difference equation. As an example, consider the nonlinear
ODE
du 1 2
+ u =0
dt
2
(6.85)
solved using implicit Euler time marching, giving
1
un+1 + h u2n+1 = un
2
(6.86)
6.10 Implementation of Implicit Methods
107
which requires a nontrivial method to solve for un+1 . There are several different approaches one can take to solving this nonlinear difference equation. An
iterative method, such as Newton’s method (see below), can be used. In practice, the “initial guess” for this nonlinear problem can be quite close to the
solution, since the “initial guess” is simply the solution at the previous time
step, which implies that a linearization approach may be quite successful.
Such an approach is described in the next section.
6.10.3 Local Linearization for Scalar Equations
General Development. Let us start the process of local linearization by
considering Eq. 6.83. In order to implement the linearization, we expand
F (u, t) about some reference point in time. Designate the reference value by
tn and the corresponding value of the dependent variable by un . A Taylor
series expansion about these reference quantities gives
µ
µ
¶
¶
∂F
∂F
F (u, t) = F (un , tn ) +
(u − un ) +
(t − tn )
∂u n
∂t n
µ
¶
µ 2 ¶
1 ∂2F
∂ F
2
+
(u − un ) +
(u − un )(t − tn )
2 ∂u2 n
∂u∂t n
µ 2 ¶
1 ∂ F
+
(t − tn )2 + · · ·
(6.87)
2 ∂t2 n
On the other hand, the expansion of u(t) in terms of the independent variable
t is
µ ¶
µ 2 ¶
∂u
1
∂ u
u(t) = un + (t − tn )
+ (t − tn )2
+ ···
(6.88)
∂t n 2
∂t2 n
If t is within h of tn , both (t − tn )k and (u − un )k are O(hk ), and Eq. 6.87
can be written
¶
µ
¶
µ
∂F
∂F
(u − un ) +
(t − tn ) + O(h2 ) (6.89)
F (u, t) = Fn +
∂u n
∂t n
Notice that this is an expansion of the derivative of the function. Thus, relative to the order of expansion of the function, it represents a second-orderaccurate, locally-linear approximation to F (u, t) that is valid in the vicinity
of the reference station tn and the corresponding un = u(tn ). With this we
obtain the locally (in the neighborhood of tn ) time-linear representation of
Eq. 6.83, namely
µ
¶
µ
µ
¶
¶ µ
¶
du
∂F
∂F
∂F
=
u + Fn −
un +
(t − tn ) + O(h2 )
dt
∂u n
∂u n
∂t n
(6.90)
108
6. TIME-MARCHING METHODS FOR ODE’S
Implementation of the Trapezoidal Method. As an example of how
such an expansion can be used, consider the mechanics of applying the trapezoidal method for the time integration of Eq. 6.83. The trapezoidal method
is given by
1
un+1 = un + h[Fn+1 + Fn ] + hO(h2 )
2
(6.91)
where we write hO(h2 ) to emphasize that the method is second order accurate. Using Eq. 6.89 to evaluate Fn+1 = F (un+1 , tn+1 ), one finds
·
µ
¶
µ
¶
¸
1
∂F
∂F
2
un+1 = un + h Fn +
(un+1 − un ) + h
+ O(h ) + Fn
2
∂u n
∂t n
+hO(h2 )
(6.92)
Note that the O(h2 ) term within the brackets (which is due to the local linearization) is multiplied by h and therefore is the same order as the hO(h2 )
error from the trapezoidal method. The use of local time linearization updated at the end of each time step and the trapezoidal time march combine
to make a second-order-accurate numerical integration process. There are,
of course, other second-order implicit time-marching methods that can be
used. The important point to be made here is that local linearization updated
at each time step has not reduced the order of accuracy of a second-order
time-marching process.
A very useful reordering of the terms in Eq. 6.92 results in the expression
·
µ
¶ ¸
µ
¶
1
∂F
1 2 ∂F
1− h
∆un = hFn + h
(6.93)
2
∂u n
2
∂t n
which is now in the delta form which will be formally introduced in Section
12.6. In many fluid mechanics applications the nonlinear function F is not an
explicit function of t. In such cases the partial derivative of F (u) with respect
to t is zero, and Eq. 6.93 simplifies to the second-order accurate expression
·
µ
¶ ¸
1
∂F
1− h
∆un = hFn
(6.94)
2
∂u n
Notice that the RHS is extremely simple. It is the product of h and the RHS
of the basic equation evaluated at the previous time step. In this example, the
basic equation was the simple scalar equation 6.83, but for our applications,
it is generally the space-differenced form of the steady-state equation of some
fluid flow problem.
A numerical time-marching procedure using Eq. 6.94 is usually implemented as follows:
~ and save ~un .
1. Solve for the elements of hF~n , store them in an array say R,
6.10 Implementation of Implicit Methods
109
2. Solve for the elements of the matrix multiplying ∆~un and store in some
appropriate manner making use of sparseness or bandedness of the matrix
if possible. Let this storage area be referred to as B.
3. Solve the coupled set of linear equations
~
B∆~un = R
for ∆~un . (Very seldom does one find B −1 in carrying out this step.)
4. Find ~un+1 by adding ∆~un to ~un , thus
~un+1 = ∆~un + ~un
The solution for ~un+1 is generally stored such that it overwrites the value
of ~un and the process is repeated.
Implementation of the Implicit Euler Method. We have seen that the
first-order implicit Euler method can be written
un+1 = un + hFn+1
(6.95)
if we introduce Eq. 6.90 into this method, rearrange terms, and remove the
explicit dependence on time, we arrive at the form
¶ ¸
·
µ
∂F
∆un = hFn
(6.96)
1−h
∂u n
We see that the only difference between the implementation of the trapezoidal
method and the implicit Euler method is the factor of 12 in the brackets of the
left side of Eqs. 6.94 and 6.96. Omission of this factor degrades the method
in time accuracy by one order of h. We shall see later that this method is an
excellent choice for steady problems.
Newton’s Method. Consider the limit h → ∞ of Eq. 6.96 obtained by
dividing both sides by h and setting 1/h = 0. There results
µ
¶
∂F
−
∆un = Fn
(6.97)
∂u n
or
·µ
un+1 = un −
∂F
∂u
¶ ¸−1
Fn
(6.98)
n
This is the well-known Newton method for finding the roots of the nonlinear
equation F (u) = 0. The fact that it has quadratic convergence is verified
by a glance at Eqs. 6.87 and 6.88 (remember the dependence on t has been
eliminated for this case). By quadratic convergence, we mean that the error
after a given iteration is proportional to the square of the error at the previous iteration, where the error is the difference between the current solution
110
6. TIME-MARCHING METHODS FOR ODE’S
and the converged solution. Quadratic convergence is thus a very powerful
property. Use of a finite value of h in Eq. 6.96 leads to linear convergence,
i.e., the error at a given iteration is some multiple of the error at the previous
iteration. The reader should ponder the meaning of letting h → ∞ for the
trapezoidal method, given by Eq. 6.94.
6.10.4 Local Linearization for Coupled Sets of Nonlinear
Equations
In order to present this concept, let us consider an example involving some
simple boundary-layer equations. We choose the Falkner-Skan equations from
classical boundary-layer theory. Our task is to apply the implicit trapezoidal
method to the equations
Ã
µ ¶2 !
d3 f
d2 f
df
=0
(6.99)
+f 2 +β 1−
dt3
dt
dt
Here f represents a dimensionless stream function, and β is a scaling factor.
First of all we reduce Eq. 6.99 to a set of first-order nonlinear equations
by the transformations
u1 =
d2 f
dt2
,
u2 =
df
dt
,
u3 = f
(6.100)
This gives the coupled set of three nonlinear equations
¡
¢
u01 = F1 = −u1 u3 − β 1 − u22
u02 = F2 = u1
u03 = F3 = u2
(6.101)
and these can be represented in vector notation as
d~u
~ (~u)
=F
dt
(6.102)
Now we seek to make the same local expansion that derived Eq. 6.90,
except that this time we are faced with a nonlinear vector function, rather
than a simple nonlinear scalar function. The required extension requires the
evaluation of a matrix, called the Jacobian matrix.9 Let us refer to this matrix
as A. It is derived from Eq. 6.102 by the following process
A = (aij ) = ∂Fi /∂uj
9
(6.103)
Recall that we derived the Jacobian matrices for the two-dimensional Euler equations in Section 2.2
6.10 Implementation of Implicit Methods
111
For the general case involving a 3 by 3 matrix this is

∂F1
 ∂u1



A =  ∂F2
 ∂u1


∂F3
∂u1

∂F1 ∂F1
∂u2 ∂u3 


∂F2 ∂F2 
∂u2 ∂u3 



∂F3 ∂F3
∂u2 ∂u3
(6.104)
~ (~u) about some reference state ~u can be expressed in
The expansion of F
n
a way similar to the scalar expansion given by eq 6.87. Omitting the explicit
~ as F
~ (~u ), one
dependency on the independent variable t, and defining F
n
n
10
has
³
´
~ (~u) = F
~ + A ~u − ~u + O(h2 )
F
(6.105)
n
n
n
where the argument for O(h2 ) is the same as in the derivation of Eq. 6.89.
Using this we can write the local linearization of Eq. 6.102 as
h
i
d~u
~ − A ~u +O(h2 )
= An~u + F
n
n n
dt
{z
}
|
(6.106)
“constant”
which is a locally-linear, second-order-accurate approximation to a set of
coupled nonlinear ordinary differential equations that is valid for t ≤ tn + h.
Any first- or second-order time-marching method, explicit or implicit, could
be used to integrate the equations without loss in accuracy with respect to
order. The number of times, and the manner in which, the terms in the
Jacobian matrix are updated as the solution proceeds depends, of course, on
the nature of the problem.
Returning to our simple boundary-layer example, which is given by Eq.
6.101, we find the Jacobian matrix to be


− u3 2βu2 −u1
0
0 
A= 1
(6.107)
0
1
0
The student should be able to derive results for this example that are equivalent to those given for the scalar case in Eq. 6.93. Thus for the Falkner-Skan
equations the trapezoidal method results in
10
The Taylor series expansion of a vector contains a vector for the first term, a
matrix times a vector for the second term, and tensor products for the terms of
higher order.
112
6. TIME-MARCHING METHODS FOR ODE’S


¡
¢ 


1+ h
(u3 )n −βh(u2 )n h
(u1 )n
− (u1 u3 )n − β 1 − u22 n
(∆u1 )n
2
2



(u1 )n
1
0  (∆u2 )n  = h 
−h

2
(∆u3 )n
(u2 )n
0
−h
1
2
We find ~un+1 from ∆~un + ~un , and the solution is now advanced one step.
Re-evaluate the elements using ~un+1 and continue. Without any iterating
within a step advance, the solution will be second-order accurate in time.
Exercises
6.1 Find an expression for the nth term in the Fibonacci series, which is given
by 1, 1, 2, 3, 5, 8, . . . Note that the series can be expressed as the solution to
a difference equation of the form un+1 = un + un−1 . What is u25 ? (Let the
first term given above be u0 .)
6.2 The trapezoidal method un+1 = un + 12 h(u0n+1 + u0n ) is used to solve the
representative ODE.
(a) What is the resulting O∆E?
(b) What is its exact solution?
(c) How does the exact steady-state solution of the O∆E compare with the
exact steady-state solution of the ODE if µ = 0?
6.3 The 2nd-order backward method is given by
un+1 =
¤
1£
4un − un−1 + 2hu0n+1
3
(a) Write the O∆E for the representative equation. Identify the polynomials
P (E) and Q(E).
(b) Derive the λ-σ relation. Solve for the σ-roots and identify them as principal or spurious.
(c) Find erλ and the first two nonvanishing terms in a Taylor series expansion
of the spurious root.
6.4 Consider the time-marching scheme given by
un+1 = un−1 +
2h 0
(u
+ u0n + u0n−1 )
3 n+1
(a) Write the O∆E for the representative equation. Identify the polynomials
P (E) and Q(E).
(b) Derive the λ − σ relation.
(c) Find erλ .
6.5 Find the difference equation which results from applying the Gazdag
predictor-corrector method (Eq. 6.65) to the representative equation. Find
the λ-σ relation.
Exercises
113
6.6 Consider the following time-marching method:
ũn+1/3 = un + hu0n /3
ūn+1/2 = un + hũ0n+1/3 /2
un+1
= un + hū0n+1/2
Find the difference equation which results from applying this method to
the representative equation. Find the λ-σ relation. Find the solution to the
difference equation, including the homogeneous and particular solutions. Find
erλ and erµ . What order is the homogeneous solution? What order is the
particular solution? Find the particular solution if the forcing term is fixed.
6.7 Write a computer program to solve the one-dimensional linear convection
equation with periodic boundary conditions and a = 1 on the domain 0 ≤
x ≤ 1. Use 2nd-order centered differences in space and a grid of 50 points.
For the initial condition, use
u(x, 0) = e−0.5[(x−0.5)/σ]
2
with σ = 0.08. Use the explicit Euler, 2nd-order Adams-Bashforth (AB2),
implicit Euler, trapezoidal, and 4th-order Runge-Kutta methods. For the
explicit Euler and AB2 methods, use a Courant number, ah/∆x, of 0.1; for the
other methods, use a Courant number of unity. Plot the solutions obtained
at t = 1 compared to the exact solution (which is identical to the initial
condition).
6.8 Repeat problem 6.7 using 4th-order (noncompact) differences in space.
Use only 4th-order Runge-Kutta time marching at a Courant number of unity.
Show solutions at t = 1 and t = 10 compared to the exact solution.
6.9 Using the computer program written for problem 6.7, compute the solution at t = 1 using 2nd-order centered differences in space coupled with the
4th-order Runge-Kutta method for grids of 100, 200, and 400 nodes. On a
log-log scale, plot the error given by
v
uM
uX (uj − uexact
)2
j
t
M
j=1
where M is the number of grid nodes and uexact is the exact solution. Find
the global order of accuracy from the plot.
6.10 Using the computer program written for problem 6.8, repeat problem
6.9 using 4th-order (noncompact) differences in space.
114
6. TIME-MARCHING METHODS FOR ODE’S
6.11 Write a computer program to solve the one-dimensional linear convection equation with inflow-outflow boundary conditions and a = 1 on the
domain 0 ≤ x ≤ 1. Let u(0, t) = sin ωt with ω = 10π. Run until a periodic steady state is reached which is independent of the initial condition
and plot your solution compared with the exact solution. Use 2nd-order centered differences in space with a 1st-order backward difference at the outflow
boundary (as in Eq. 3.69) together with 4th-order Runge-Kutta time marching. Use grids with 100, 200, and 400 nodes and plot the error vs. the number
of grid nodes, as described in problem 6.9. Find the global order of accuracy.
6.11 Repeat problem 6.10 using 4th-order (noncompact) centered differences. Use a third-order forward-biased operator at the inflow boundary (as
in Eq. 3.67). At the last grid node, derive and use a 3rd-order backward operator (using nodes j − 3, j − 2, j − 1, and j) and at the second last node,
use a 3rd-order backward-biased operator (using nodes j − 2, j − 1, j, and
j + 1; see problem 3.1 in Chapter 3).
6.13 Using the approach described in Section 6.6.2, find the phase speed
error, erp , and the amplitude error, era , for the combination of second-order
centered differences and 1st, 2nd, 3rd, and 4th-order Runge-Kutta timemarching at a Courant number of unity. Also plot the phase speed error
obtained using exact integration in time, i.e., that obtained using the spatial
discretization alone. Note that the required σ-roots for the various RungeKutta methods can be deduced from Eq. 6.69, without actually deriving the
methods. Explain your results.