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Page 1: Fundamentals of Computational Fluid Dynamics · PDF filethis document downloaded from... chet-aero.com the wings of flight in an age of adventure Use of this document subject to the

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of adventure

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Page 2: Fundamentals of Computational Fluid Dynamics · PDF filethis document downloaded from... chet-aero.com the wings of flight in an age of adventure Use of this document subject to the

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6. TIME-MARCHING METHODS FORODE’S

After discretizing the spatial derivatives in the governing PDE’s (such as theNavier-Stokes equations), we obtain a coupled system of nonlinear ODE’s inthe form

d~u

dt= ~F (~u, t) (6.1)

These can be integrated in time using a time-marching method to obtain atime-accurate solution to an unsteady flow problem. For a steady flow prob-lem, spatial discretization leads to a coupled system of nonlinear algebraicequations in the form

~F (~u) = 0 (6.2)

As a result of the nonlinearity of these equations, some sort of iterativemethod is required to obtain a solution. For example, one can consider the useof Newton’s method, which is widely used for nonlinear algebraic equations(see Section 6.10.3). This produces an iterative method in which a coupledsystem of linear algebraic equations must be solved at each iteration. Thesecan be solved iteratively using relaxation methods, which will be discussed inChapter 9, or directly using Gaussian elimination or some variation thereof.

Alternatively, one can consider a time-dependent path to the steady stateand use a time-marching method to integrate the unsteady form of the equa-tions until the solution is sufficiently close to the steady solution. The subjectof the present chapter, time-marching methods for ODE’s, is thus relevantto both steady and unsteady flow problems. When using a time-marchingmethod to compute steady flows, the goal is simply to remove the transientportion of the solution as quickly as possible; time-accuracy is not required.This motivates the study of stability and stiffness, topics which are coveredin the next two chapters.

Application of a spatial discretization to a PDE produces a coupled systemof ODE’s. Application of a time-marching method to an ODE produces anordinary difference equation (O∆E ). In earlier chapters, we developed exactsolutions to our model PDE’s and ODE’s. In this chapter we will present somebasic 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’sarising from the application of time-marching methods to the model ODE’s.

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82 6. TIME-MARCHING METHODS FOR ODE’S

6.1 Notation

Using the semi-discrete approach, we reduce our PDE to a set of coupledODE’s represented in general by Eq. 4.1. However, for the purpose of thischapter, we need only consider the scalar case

du

dt= u′ = F (u, t) (6.3)

Although we use u to represent the dependent variable, rather than w, thereader should recall the arguments made in Chapter 4 to justify the study ofa scalar ODE. Our first task is to find numerical approximations that can beused to carry out the time integration of Eq. 6.3 to some given accuracy, whereaccuracy can be measured either in a local or a global sense. We then face afurther task concerning the numerical stability of the resulting methods, butwe 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 thetime interval ∆t. Combining this notation with Eq. 6.3 gives

u′n = Fn = F (un, tn) , tn = nh

Often we need a more sophisticated notation for intermediate time stepsinvolving temporary calculations denoted by u, u, etc. For these we use thenotation

u′n+α = Fn+α = F (un+α, tn + αh)

The choice of u′ or F to express the derivative in a scheme is arbitrary. Theyare both commonly used in the literature on ODE’s.

The methods we study are to be applied to linear or nonlinear ODE’s, butthe methods themselves are formed by linear combinations of the dependentvariable and its derivative at various time intervals. They are representedconceptually by

un+1 = f(β1u

′n+1, β0u

′n, β−1u

′n−1, · · · , α0un, α−1un−1, · · ·

)(6.4)

With an appropriate choice of the α’s and β’s, these methods can be con-structed to give a local Taylor series accuracy of any order. The methods aresaid to be explicit if β1 = 0 and implicit otherwise. An explicit method isone in which the new predicted solution is only a function of known data,for example, u′n, u′n−1, un, and un−1 for a method using two previous timelevels, and therefore the time advance is simple. For an implicit method, thenew predicted solution is also a function of the time derivative at the newtime level, that is, u′n+1. As we shall see, for systems of ODE’s and nonlinearproblems, implicit methods require more complicated strategies to solve forun+1 than explicit methods.

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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-marchinggeneral ODE’s are:

un+1 = un + hu′n (6.5)

un+1 = un + hu′n+1 (6.6)

and

un+1 = un + hu′n

un+1 =12[un + un+1 + hu′n+1] (6.7)

According to the conditions presented under Eq. 6.4, the first and third ofthese are examples of explicit methods. We refer to them as the explicit Eulermethod and the MacCormack predictor-corrector method,1 respectively. Thesecond is implicit and referred to as the implicit (or backward) Euler method.

These methods are simple recipes for the time advance of a function interms of its value and the value of its derivative, at given time intervals. Thematerial presented in Chapter 4 develops a basis for evaluating such methodsby introducing the concept of the representative equation

du

dt= u′ = λu + aeµt (6.8)

written here in terms of the dependent variable, u. The value of this equa-tion arises from the fact that, by applying a time-marching method, we cananalytically convert such a linear ODE into a linear O∆E . The latter aresubject to a whole body of analysis that is similar in many respects to, andjust as powerful as, the theory of ODE’s. We next consider examples of thisconversion 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 thedependent variable un and the independent variable n. As another example,applying the implicit Euler method, Eq. 6.6, to Eq. 6.8, we find1 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 bepresented in Section 11.3

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84 6. TIME-MARCHING METHODS FOR ODE’S

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

un+1 − (1 + λh)un = aheµhn

−12(1 + λh)un+1 + un+1 − 1

2un =

12aheµh(n+1) (6.11)

which is a coupled set of linear O∆E’s with constant coefficients. Note thatthe first line in Eq. 6.11 is identical to Eq. 6.9, since the predictor step inEq. 6.7 is simply the explicit Euler method. The second line in Eq. 6.11 isobtained by noting that

u′n+1 = F (un+1, tn + h)

= λun+1 + aeµh(n+1) (6.12)

Now we need to develop techniques for analyzing these difference equa-tions so that we can compare the merits of the time-marching methods thatgenerated them.

6.3 Solution of Linear O∆E’s With ConstantCoefficients

The techniques for solving linear difference equations with constant coef-ficients is as well developed as that for ODE’s, and the theory follows aremarkably parallel path. This is demonstrated by repeating some of thedevelopments in Section 4.2, but for difference rather than differential equa-tions.

6.3.1 First- and Second-Order Difference Equations

First-Order Equations. The simplest nonhomogeneous O∆E of interest isgiven by the single first-order equation

un+1 = σun + abn (6.13)

where σ, a, and b are, in general, complex parameters. The independentvariable is now n rather than t, and since the equations are linear and haveconstant coefficients, σ is not a function of either n or u. The exact solutionof Eq. 6.13 is

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6.3 Solution of Linear O∆E’s With Constant Coefficients 85

un = c1σn +

abn

b− σ

where c1 is a constant determined by the initial conditions. In terms of theinitial value of u it can be written

un = u0σn + a

bn − σn

b− σ

Just as in the development of Eq. 4.10, one can readily show that the solutionof the defective case, (b = σ),

un+1 = σun + aσn

isun =

[u0 + anσ−1

]σn

This can all be easily verified by substitution.Second-Order Equations. The homogeneous form of a second-order dif-ference equation is given by

un+2 + a1un+1 + a0un = 0 (6.14)

Instead of the differential operator D ≡ ddt used for ODE’s, we use for O∆E’s

the difference operator E (commonly referred to as the displacement or shiftoperator) and defined formally by the relations

un+1 = Eun , un+k = Ekun

Further notice that the displacement operator also applies to exponents, thus

bα · bn = bn+α = Eα · 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 intro-

duced 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

(E2 + a1E + 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) whichplays the same role for difference equations that P (D) played for differentialequations; that is, its roots determine the solution to the O∆E. In the analysisof O∆E’s, we label these roots σ1, σ2, · · ·, etc, and refer to them as the σ-roots. They are found by solving the equation P (σ) = 0. In the simple examplegiven above, there are just two σ roots, and in terms of them the solutioncan 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 isa solution to Eq. 6.14 for all c1, c2, and n should be verified by substitution.

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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 differenceequations have the form

u(n+1)1 = c11u

(n)1 + c12u

(n)2

u(n+1)2 = c21u

(n)1 + c22u

(n)2 (6.17)

which can also be written

~un+1 = C~un , ~un =[u

(n)1 , u

(n)2

]T

, C =[

c11 c12

c21 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 characteristicpolynomial, this time having the form P (E) = det [C − E I ]. The σ-rootsare found from

P (σ) = det[

(c11 − σ) c12

c21 (c22 − σ)

]= 0

Obviously, the σk are the eigenvalues of C, and following the logic ofSection 4.2, if ~x are its eigenvectors, the solution of Eq. 6.17 is

~un =2∑

k=1

ck(σk)n~xk

where ck are constants determined by the initial conditions.

A Defective System. The solution of O∆E’s with defective eigensystemsfollows closely the logic in Section 4.2.2 for defective ODE’s. For example,one can show that the solution to

un+1

un+1

un+1

=

σ1 σ

1 σ

un

un

un

is

un = u0σn

un =[u0 + u0nσ−1

]σn

un =[u0 + u0nσ−1 + u0

n(n− 1)2

σ−2

]σn (6.18)

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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 equa-tions produced by applying a time-marching method to the representativeequation 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 − 1

2

] [uu

]

n

= h ·[

112E

]· aeµhn (6.21)

All three of these equations are subsets of the operational form of therepresentative O∆E

P (E)un = Q(E) · aeµhn (6.22)

which is produced by applying time-marching methods to the representativeODE, Eq. 4.45. We can express in terms of Eq. 6.22 all manner of stan-dard time-marching methods having multiple time steps and various typesof intermediate predictor-corrector families. The terms P (E) and Q(E) arepolynomials in E referred to as the characteristic polynomial and the partic-ular polynomial, respectively.

The general solution of Eq. 6.22 can be expressed as

un =K∑

k=1

ck(σk)n + aeµhn · Q(eµh)P (eµh)

(6.23)

where σk are the K roots of the characteristic polynomial, P (σ) = 0. Whendeterminants are involved in the construction of P (E) and Q(E), as wouldbe 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 (twoin this case) solutions, one for un and one for un, and we are usually onlyinterested in the final solution un. Notice also, the important subset of thissolution which occurs when µ = 0, representing a time-invariant particularsolution, or a steady state. In such a case

un =K∑

k=1

ck(σk)n + a · Q(1)P (1)

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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 · h

eµh − 1− λh

For the implicit Euler method, Eq. 6.20, we have

P (E) = (1− λh)E − 1Q(E) = hE (6.25)

so

un = c1

(1

1− λh

)n

+ aeµhn · heµh

(1− λh)eµh − 1

In the case of the coupled predictor-corrector equations, Eq. 6.21, one solvesfor the final family un (one can also find a solution for the intermediate familyu), and there results

P (E) = det[

E −(1 + λh)− 1

2(1 + λh)E E − 12

]= E

(E − 1− λh− 1

2λ2h2

)

Q(E) = det[

E h− 1

2(1 + λh)E 12hE

]=

12hE(E + 1 + λh)

The σ-root is found from

P (σ) = σ

(σ − 1− λh− 1

2λ2h2

)= 0

which has only one nontrivial root (σ = 0 is simply a shift in the referenceindex). The complete solution can therefore be written

un = c1

(1 + λh +

12λ2h2

)n

+ aeµhn ·12h(eµh + 1 + λh

)

eµh − 1− λh− 12λ2h2

(6.26)

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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 andthe σ-roots. The former are the eigenvalues of the A matrix in the ODE’sfound by space differencing the original PDE, and the latter are the roots ofthe characteristic polynomial in a representative O∆E found by applying atime-marching method to the representative ODE. There is a fundamentalrelation between the two which can be used to identify many of the essentialproperties of a time-march method. This relation is first demonstrated bydeveloping it for the explicit Euler method.

First we make use of the semi-discrete approach to find a system of ODE’sand then express its solution in the form of Eq. 4.27. Remembering thatt = nh, one can write

~u(t) = c1

(eλ1h

)n ~x1 + · · ·+ cm

(eλmh

)n ~xm + · · ·+ cM

(eλM h

)n ~xM + P.S.

(6.27)

where for the present we are not interested in the form of the particularsolution (P.S.). Now the explicit Euler method produces for each λ-root, oneσ-root, which is given by σ = 1 + λh. So if we use the Euler method for thetime advance of the ODE’s, the solution2 of the resulting O∆E is

~un = c1(σ1)n ~x1 + · · ·+ cm(σm)n ~xm + · · ·+ cM (σM )n ~xM + P.S.

(6.28)

where the cm and the ~xm in the two equations are identical and σm =(1 + λmh). Comparing Eq. 6.27 and Eq. 6.28, we see a correspondence be-tween σm and eλmh. Since the value of eλh can be expressed in terms of theseries

eλh = 1 + λh +12λ2h2 +

16λ3h3 + · · ·+ 1

n!λnhn + · · ·

the truncated expansion σ = 1+λh is a reasonable3 approximation for smallenough λh.

Suppose, instead of the Euler method, we use the leapfrog method for thetime advance, which is defined by

un+1 = un−1 + 2hu′n (6.29)

Applying Eq. 6.29 to Eq. 6.8, we have the characteristic polynomial P (E) =E2 − 2λhE − 1, so that for every λ the σ must satisfy the relation

σ2m − 2λmhσm − 1 = 0 (6.30)

2 Based on Section 4.4.3 The error is O(λ2h2).

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90 6. TIME-MARCHING METHODS FOR ODE’S

Now we notice that each λ produces two σ-roots. For one of these we find

σm = λmh +√

1 + λ2mh2 (6.31)

= 1 + λmh +12λ2

mh2 − 18λ4

mh4 + · · · (6.32)

This is an approximation to eλmh with an error O(λ3h3). The other root,λmh −

√1 + λ2

mh2, will be discussed in Section 6.5.3.

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 ofthe equations in a coupled system of linear ODE’s in theform of Eq. 4.6 always produces one σ-root for every λ-rootthat satisfies the relation

σ = 1 + λh +12λ2h2 + · · ·+ 1

k!λkhk + O

(hk+1

)

where k is the order of the time-marching method.

(6.33)

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 thedetails of the time-marching method, knowing only that its leading error isO

(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

(δtu)n =12h

(un+1 − un−1) (6.34)

has a leading truncation error which is O(h2), while the second-order time-marching method which results from this approximation, which is the leapfrogmethod:

un+1 = un−1 + 2hu′n (6.35)

has a leading truncation error O(h3). This arises simply because of our nota-tion for the time-marching method in which we have multiplied through byh to get an approximation for the function un+1 rather than the derivativeas in Eq. 6.34. The following example makes this clear. Consider a solutionobtained at a given time T using a second-order time-marching method witha time step h. Now consider the solution obtained using the same methodwith a time step h/2. Since the error per time step is O(h3), this is reducedby a factor of eight (considering the leading term only). However, twice asmany time steps are required to reach the time T . Therefore the error atthe end of the simulation is reduced by a factor of four, consistent with asecond-order approximation.

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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 producestwo σ-roots for each λ. One of these we identified as the principal root, whichalways has the property given in 6.33. The other is referred to as a spuriousσ-root and designated (σm)2. In general, the λ − σ relation produced by atime-marching scheme can result in multiple σ-roots all of which, except forthe principal one, are spurious. All spurious roots are designated (σm)k wherek = 2, 3, · · ·. No matter whether a σ-root is principal or spurious, it is alwayssome algebraic function of the product λh. To express this fact we use thenotation σ = σ(λh).

If a time-marching method produces spurious σ-roots, the solution for theO∆E in the form shown in Eq. 6.28 must be modified. Following again themessage of Section 4.4, we have

~un = c11(σ1)n1

~x1 + · · ·+ cm1(σm)n1

~xm + · · ·+ cM1(σM )n1

~xM + P.S.

+c12(σ1)n2

~x1 + · · ·+ cm2(σm)n2

~xm + · · ·+ cM2(σM )n2

~xM

+c13(σ1)n3

~x1 + · · ·+ cm3(σm)n3

~xm + · · ·+ cM3(σM )n3

~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 toadvance the solution from time level n to n + 1. Such roots originate entirelyfrom the numerical approximation of the time-marching method and havenothing to do with the ODE being solved. However, generation of spuriousroots does not, in itself, make a method inferior. In fact, many very accuratemethods in practical use for integrating some forms of ODE’s have spuriousroots.

It should be mentioned that methods with spurious roots are not self-starting. For example, if there is one spurious root to a method, all of thecoefficients (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 thecoefficients. This results because methods which produce spurious roots re-quire data from time level n− 1 or earlier. For example, the leapfrog methodrequires ~un−1 and thus cannot be started using only ~un.

Presumably (i.e., if one starts the method properly4) the spurious coef-ficients are all initialized with very small magnitudes, and presumably themagnitudes of the spurious roots themselves are all less than one (see Chap-ter 7). Then the presence of spurious roots does not contaminate the answer.That is, after some finite time, the amplitude of the error associated withthe spurious roots is even smaller then when it was initialized. Thus whilespurious roots must be considered in stability analysis, they play virtually norole in accuracy analysis.4 Normally using a self-starting method for the first step or steps, as required.

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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-rootfor each λ-root. We refer to them as one-root methods. They are also calledone-step methods. They have the significant advantage of being self-startingwhich carries with it the very useful property that the time-step interval canbe changed at will throughout the marching process. Three one-root methodswere analyzed in Section 6.4.2. A popular method having this property, theso-called θ-method, is given by the formula

un+1 = un + h[(1− θ)u′n + θu′n+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 (witha complete eigensystem) having time-invariant forcing terms with the exactsolution to the O∆E’s for one-root methods. These are

~u(t) = c1

(eλ1h

)n ~x1 + · · ·+ cm

(eλmh

)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, sothat all the constants, vectors, and matrices are identical except the ~u andthe terms inside the parentheses on the right hand sides. The only error madeby introducing the time marching is the error that σ makes in approximatingeλh.

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 andevaluate 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 (seeSection 3.4). It is usually used as the basis for establishing the order of amethod. The other is the error determined at the end of a given event whichhas covered a specific interval of time composed of many time steps. This isa global error. It is useful for comparing methods, as we shall see in Chapter8.

It is quite common to judge a time-marching method on the basis of resultsfound from a Taylor table. However, a Taylor series analysis is a very limitedtool for finding the more subtle properties of a numerical time-marchingmethod. For example, it is of no use in:

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6.6 Accuracy Measures of Time-Marching Methods 93

• finding spurious roots.• evaluating numerical stability and separating the errors in phase and am-

plitude.• 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 wemake use of the material developed in the previous sections of this chapter.Our error measures are based on the difference between the exact solution tothe representative ODE, given by

u(t) = ceλt +aeµt

µ− λ(6.38)

and the solution to the representative O∆E’s, including only the contributionfrom the principal root, which can be written as

un = c1(σ1)n + 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 orglobal, is to some extent arbitrary. However, a necessary condition for thechoice 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 methodsproduce 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 isthe value of the difference between solutions based on these two roots. Wedesignate this by erλ and make the following definition

erλ ≡ eλh − σ1

The leading error term can be found by expanding in a Taylor series andchoosing the first nonvanishing term. This is similar to the error found froma Taylor table. The order of the method is the last power of λh matchedexactly.

Amplitude and Phase Error. Suppose a λ eigenvalue is imaginary. Suchcan indeed be the case when we study the equations governing periodic con-vection, which produces harmonic motion. For such cases it is more meaning-ful to express the error in terms of amplitude and phase. Let λ = iω where ωis a real number representing a frequency. Then the numerical method mustproduce a principal σ-root that is complex and expressible in the form

σ1 = σr + iσi ≈ eiωh (6.40)

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94 6. TIME-MARCHING METHODS FOR ODE’S

From this it follows that the local error in amplitude is measured by thedeviation of |σ1| from unity, that is

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 time-marching methods for convection and wave propagation phenomena.

The approach to error analysis described in Section 3.5 can be extendedto the combination of a spatial discretization and a time-marching methodapplied to the linear convection equation. The principal root, σ1(λh), isfound using λ = −iaκ∗, where κ∗ is the modified wavenumber of the spa-tial discretization. Introducing the Courant number, Cn = ah/∆x, we haveλh = −iCnκ∗∆x. Thus one can obtain values of the principal root over therange 0 ≤ κ∆x ≤ π for a given value of the Courant number. The aboveexpression for erω can be normalized to give the error in the phase speed, asfollows

erp =erω

ωh= 1 +

tan−1 [(σ1)i/(σ1)r)]Cnκ∆x

(6.42)

where ω = −aκ. A positive value of erp corresponds to phase lag (the numer-ical phase speed is too small), while a negative value corresponds to phaselead (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 theparticular solution of the ODE with that for the O∆E. We have found theseto be given by

P.S.(ODE) = aeµt · 1(µ− λ)

and

P.S.(O∆E) = aeµt · Q(eµh)P (eµh)

respectively. For a measure of the local error in the particular solution weintroduce the definition

erµ ≡ h

{P.S.(O∆E)

P.S.(ODE)− 1

}(6.43)

The multiplication by h converts the error from a global measure to a localone, so that the order of erλ and erµ are consistent. In order to determine

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6.6 Accuracy Measures of Time-Marching Methods 95

the leading error term, Eq. 6.43 can be written in terms of the characteristicand particular polynomials as

erµ =co

µ− λ· {(µ− λ)Q

(eµh

)− P(eµh

)}(6.44)

and expanded in a Taylor series, where

co = limh→0

h(µ− λ)P

(eµh

)

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 thiscase, 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 atime-marching method by the process that has been outlined. An illustrationof 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 thenonlinear cases as well as for the linear ones. A necessary condition for thisto occur is that the local accuracies of both the transient and the particularsolutions be of the same order. More precisely, a time-marching method issaid to be of order k if

erλ = c1 · (λh)k1+1 (6.45)erµ = c2 · (λh)k2+1 (6.46)where k = smallest of(k1, k2) (6.47)

The reader should be aware that this is not sufficient. For example, to deriveall of the necessary conditions for the fourth-order Runge-Kutta method pre-sented later in this chapter the derivation must be performed for a nonlinearODE. However, the analysis based on a linear nonhomogeneous ODE pro-duces the appropriate conditions for the majority of time-marching methodsused in CFD.

6.6.5 Global Accuracy

In contrast to the local error measures which have just been discussed, wecan also define global error measures. These are useful when we come to theevaluation of time-marching methods for specific purposes. This subject iscovered in Chapter 8 after our introduction to stability in Chapter 7.

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96 6. TIME-MARCHING METHODS FOR ODE’S

Suppose we wish to compute some time-accurate phenomenon over a fixedinterval of time using a constant time step. We refer to such a computationas an “event”. Let T be the fixed time of the event and h be the chosen stepsize. 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 theerror 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 aremore concerned with the global error in amplitude and phase. These are givenby

Era = 1−(√

(σ1)2r + (σ1)2i

)N

(6.49)

and

Erω ≡ N

[ωh− tan−1

((σ1)i

(σ1)r

)]

= ωT −N tan−1 [(σ1)i/(σ1)r] (6.50)

Global error in the particular solution. Finally, the global error in theparticular solution follows naturally by comparing the solutions to the ODEand the O∆E. It can be measured by

Erµ ≡ (µ− λ)Q

(eµh

)

P(eµh

) − 1

6.7 Linear Multistep Methods

In the previous sections, we have developed the framework of error analy-sis for time advance methods and have randomly introduced a few methodswithout addressing motivational, developmental, or design issues. In the sub-sequent sections, we introduce classes of methods along with their associatederror analysis. We shall not spend much time on development or design ofthese methods, since most of them have historic origins from a wide varietyof disciplines and applications. The Linear Multistep Methods (LMM’s) areprobably the most natural extension to time marching of the space differenc-ing schemes introduced in Chapter 3 and can be analyzed for accuracy ordesigned using the Taylor table approach of Section 3.4.

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6.7 Linear Multistep Methods 97

6.7.1 The General Formulation

When applied to the nonlinear ODE

du

dt= u′ = F (u, t)

all linear multistep methods can be expressed in the general form

1∑

k=1−K

αkun+k = h

1∑

k=1−K

βkFn+k (6.51)

where the notation for F is defined in Section 6.1. The methods are said tobe linear because the α’s and β’s are independent of u and n, and they aresaid to be K-step because K time-levels of data are required to march thesolution 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 theresult is expressed in operational form, one finds

(1∑

k=1−K

αkEk

)un = h

(1∑

k=1−K

βkEk

)(λun + aeµhn) (6.52)

We recall from Section 6.5.2 that a time-marching method when applied tothe representative equation must provide a σ-root, labeled σ1, that approx-imates eλh through the order of the method. The condition referred to asconsistency simply means that σ → 1 as h → 0, and it is certainly a neces-sary condition for the accuracy of any time-marching method. We can alsoagree that, to be of any value in time accuracy, a method should at least befirst-order accurate, that is σ → (1 + λh) as h → 0. One can show that theseconditions are met by any method represented by Eq. 6.51 if

k

αk = 0 and∑

k

βk =∑

k

(K + k − 1)αk

Since both sides of Eq. 6.51 can be multiplied by an arbitrary constant, thesemethods are often “normalized” by requiring

k

βk = 1

Under this condition, co = 1 in Eq. 6.44.

6.7.2 Examples

There are many special explicit and implicit forms of linear multistep meth-ods. Two well-known families of them, referred to as Adams-Bashforth (ex-plicit) and Adams-Moulton (implicit), can be designed using the Taylor table

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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 constraintthat β1 = 0. The three-step Adams-Moulton method can be written in thefollowing form

un+1 = un + h(β1u′n+1 + β0u

′n + β−1u

′n−1 + β−2u

′n−2) (6.54)

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 12 0 −2 −43 0 3 124 0 −4 −32

β1

β0

β−1

β−2

=

1111

(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 oneobtains

1 1 10 −2 −40 3 12

β0

β−1

β−2

=

111

(6.57)

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 appli-

cations, is given below together with identifying names that are sometimesassociated with them. In the following material AB(n) and AM(n) are used asabbreviations for the (n)th order Adams-Bashforth and (n)th order Adams-Moulton methods. One can verify that the Adams type schemes given belowsatisfy Eqs. 6.55 and 6.57 up to the order of the method.Explicit Methods.

un+1 = un + hu′n Eulerun+1 = un−1 + 2hu′n Leapfrogun+1 = un + 1

2h[3u′n − u′n−1

]AB2

un+1 = un + h12

[23u′n − 16u′n−1 + 5u′n−2

]AB3

5 Recall from Section 6.5.2 that a kth-order time-marching method has a leadingtruncation error term which is O(hk+1).

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6.7 Linear Multistep Methods 99

un

h·u′ n

h2·u′′ n

h3·u′′′ n

h4·u′′′′

n

un+

11

11 2

1 61 24

−u

n−1

−hβ

1u′ n+

1−β

1−β

1−β

11 2

−β11 6

−hβ

0u′ n

−β0

−hβ−

1u′ n−

1−β

−1

β−

1−β

−11 2

β−

11 6

−hβ−

2u′ n−

2−(−2

)0β−

2−(−2

)1β−

2−(−2

)2β−

21 2

−(−2

)3β−

21 6

Tab

le6.

1.Tay

lor

tabl

efo

rth

eA

dam

s-M

oult

onth

ree-

step

linea

rm

ulti

step

met

hod.

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100 6. TIME-MARCHING METHODS FOR ODE’S

Implicit Methods.

un+1 = un + hu′n+1 Implicit Eulerun+1 = un + 1

2h[u′n + u′n+1

]Trapezoidal (AM2)

un+1 = 13[4un − un−1 + 2hu′n+1

]2nd-order Backward

un+1 = un + h12

[5u′n+1 + 8u′n − u′n−1

]AM3

6.7.3 Two-Step Linear Multistep Methods

High-resolution CFD problems usually require very large data sets to storethe spatial information from which the time derivative is calculated. Thislimits the interest in multistep methods to about two time levels. The mostgeneral two-step linear multistep method (i.e., K = 2 in Eq. 6.51) that is atleast first-order accurate can be written as

(1 + ξ)un+1 = [(1 + 2ξ)un − ξun−1] + h [ θu′n+1 + (1− θ + ϕ)u′n − ϕu′n−1

]

(6.59)

Clearly the methods are explicit if θ = 0 and implicit otherwise. A list ofmethods contained in Eq. 6.59 is given in Table 6.2. Notice that the Adamsmethods have ξ = 0, which corresponds to α−1 = 0 in Eq. 6.51. Methodswith ξ = −1/2, which corresponds to α0 = 0 in Eq. 6.51, are known as Milnemethods.

θ ξ ϕ Method Order

0 0 0 Euler 11 0 0 Implicit Euler 1

1/2 0 0 Trapezoidal or AM2 21 1/2 0 2nd Order Backward 2

3/4 0 −1/4 Adams type 21/3 −1/2 −1/3 Lees 21/2 −1/2 −1/2 Two–step trapezoidal 25/9 −1/6 −2/9 A–contractive 2

0 −1/2 0 Leapfrog 20 0 1/2 AB2 20 −5/6 −1/3 Most accurate explicit 3

1/3 −1/6 0 Third–order implicit 35/12 0 1/12 AM3 31/6 −1/2 −1/6 Milne 4

Table 6.2. Some linear one- and two-step methods; see Eq. 6.59.

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6.8 Predictor-Corrector Methods 101

One can show after a little algebra that both erµ and erλ are reduced to0(h3) (i.e., the methods are 2nd-order accurate) if

ϕ = ξ − θ +12

The class of all 3rd-order methods is determined by imposing the additionalconstraint

ξ = 2θ − 56

Finally a unique fourth-order method is found by setting θ = −ϕ = −ξ/3 =16.

6.8 Predictor-Corrector Methods

There are a wide variety of predictor-corrector schemes created and used for avariety of purposes. Their use in solving ODE’s is relatively easy to illustrateand understand. Their use in solving PDE’s can be much more subtle anddemands concepts6 which have no counterpart in the analysis of ODE’s.

Predictor-corrector methods constructed to time-march linear or nonlinearODE’s are composed of sequences of linear multistep methods, each of whichis referred to as a family in the solution process. There may be many familiesin the sequence, and usually the final family has a higher Taylor-series orderof accuracy than the intermediate ones. Their use is motivated by ease ofapplication and increased efficiency, where measures of efficiency are discussedin the next two chapters.

A simple one-predictor, one-corrector example is given by

un+α = un + αhu′nun+1 = un + h

[βu′n+α + γu′n

](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 equationand using the operational techniques outlined in Section 6.4. It is easy toshow, following the example leading to Eq. 6.26, that

P (E) = Eα · [E − 1− (γ + β)λh− αβλ2h2]

(6.61)Q(E) = Eα · h · [βEα + γ + αβλh] (6.62)

Considering only local accuracy, one is led, by following the discussion inSection 6.6, to the following observations. For the method to be second-orderaccurate both erλ and erµ must be O(h3). For this to hold for erλ, it is obviousfrom Eq. 6.61 that6 Such as alternating direction, fractional-step, and hybrid methods.

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102 6. TIME-MARCHING METHODS FOR ODE’S

γ + β = 1 ; αβ =12

which provides two equations for three unknowns. The situation for erµ re-quires some algebra, but it is not difficult to show using Eq. 6.44 that the sameconditions also make it O(h3). One concludes, therefore, that the predictor-corrector sequence

un+α = un + αhu′n

un+1 = un +12h

[(1α

)u′n+α +

(2α− 1

α

)u′n

](6.63)

is a second-order accurate method for any α.A classical predictor-corrector sequence is formed by following an Adams-

Bashforth predictor of any order with an Adams-Moulton corrector having anorder one higher. The order of the combination is then equal to the order ofthe 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

un+1 = un +12h[3u′n − u′n−1

]

un+1 = un +h

12[5u′n+1 + 8u′n − u′n−1

](6.64)

Some simple, specific, second-order accurate methods are given below. TheGazdag method, which we discuss in Chapter 8, is

un+1 = un +12h[3u′n − u′n−1

]

un+1 = un +12h[u′n + u′n+1

](6.65)

The Burstein method, obtained from Eq. 6.63 with α = 1/2 is

un+1/2 = un +12hu′n

un+1 = un + hu′n+1/2 (6.66)

and, finally, MacCormack’s method, presented earlier in this chapter, is

un+1 = un + hu′n

un+1 =12[un + un+1 + hu′n+1] (6.67)

Note that MacCormack’s method can also be written as

un+1 = un + hu′n

un+1 = un +12h[u′n + u′n+1] (6.68)

from which it is clear that it is obtained from Eq. 6.63 with α = 1.

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6.9 Runge-Kutta Methods 103

6.9 Runge-Kutta Methods

There is a special subset of predictor-corrector methods, referred to as Runge-Kutta 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 ofthe 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

σ = 1 + λh +12λ2h2 + · · ·+ 1

k!λkhk (6.69)

It is not particularly difficult to build this property into a method, but, aswe pointed out in Section 6.6.4, it is not sufficient to guarantee kth orderaccuracy for the solution of u′ = F (u, t) or for the representative equation.To ensure kth order accuracy, the method must further satisfy the constraintthat

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 usuallyintroduced in the form

k1 = hF (un, tn)k2 = hF (un + βk1, tn + αh)k3 = hF (un + β1k1 + γ1k2, tn + α1h)k4 = hF (un + β2k1 + γ2k2 + δ2k3, tn + α2h)

followed by

u(tn + h)− u(tn) = µ1k1 + µ2k2 + µ3k3 + µ4k4 (6.71)

However, we prefer to present it using predictor-corrector notation. Thus, ascheme entirely equivalent to 6.71 is

un+α = un + βhu′nun+α1 = un + β1hu′n + γ1hu′n+α

un+α2 = un + β2hu′n + γ2hu′n+α + δ2hu′n+α1

un+1 = un + µ1hu′n + µ2hu′n+α + µ3hu′n+α1+ µ4hu′n+α2

(6.72)

Appearing in Eqs. 6.71 and 6.72 are a total of 13 parameters which are tobe determined such that the method is fourth-order according to the require-ments 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 relations7 Although implicit and multi-step Runge-Kutta methods exist, we will consider

only single-step, explicit Runge-Kutta methods here.

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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 pa-rameters 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 (1)µ2α + µ3α1 + µ4α2 = 1/2 (2)

µ3αγ1 + µ4(αγ2 + α1δ2) = 1/6 (3)µ4αγ1δ2 = 1/24 (4)

(6.74)

These four relations guarantee that the five terms in σ exactly match the first5 terms in the expansion of eλh . To satisfy the condition that erµ = O(h5),we have to fulfill four more conditions

µ2α2 + µ3α

21 + µ4α

22 = 1/3 (3)

µ2α3 + µ3α

31 + µ4α

32 = 1/4 (4)

µ3α2γ1 + µ4(α2γ2 + α2

1δ2) = 1/12 (4)µ3αα1γ1 + µ4α2(αγ2 + α1δ2) = 1/8 (4)

(6.75)

The number in parentheses at the end of each equation indicates the orderthat is the basis for the equation. Thus if the first 3 equations in 6.74 and thefirst equation in 6.75 are all satisfied, the resulting method would be third-order accurate. As discussed in Section 6.6.4, the fourth condition in Eq. 6.75cannot be derived using the methodology presented here, which is based on alinear nonhomogenous representative ODE. A more general derivation basedon 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 the10 unknowns. Since the equations are underdetermined, two parameters canbe set arbitrarily. Several choices for the parameters have been proposed, butthe most popular one is due to Runge. It results in the “standard” fourth-order Runge-Kutta method expressed in predictor-corrector form as

un+1/2 = un +12hu′n

un+1/2 = un +12hu′n+1/2

un+1 = un + hu′n+1/2

un+1 = un +16h[u′n + 2

(u′n+1/2 + u′n+1/2

)+ u′n+1

](6.76)

8 The present approach based on a linear nonhomogeneous equation provides allof the necessary conditions for Runge-Kutta methods of up to third order.

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6.10 Implementation of Implicit Methods 105

Notice that this represents the simple sequence of conventional linear multi-step methods referred to, respectively, as

Euler PredictorEuler CorrectorLeapfrog PredictorMilne Corrector

≡ RK4

One can easily show that both the Burstein and the MacCormack methodsgiven by Eqs. 6.66 and 6.67 are second-order Runge-Kutta methods, andthird-order methods can be derived from Eqs. 6.72 by setting µ4 = 0 andsatisfying 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 requirek evaluations of the derivative function to advance the solution one timestep. We shall discuss the consequences of this in Chapter 8. Higher-orderRunge-Kutta methods can be developed, but they require more derivativeevaluations than their order. For example, a fifth-order method requires sixevaluations to advance the solution one step. In any event, storage require-ments reduce the usefulness of Runge-Kutta methods of order higher thanfour for CFD applications.

6.10 Implementation of Implicit Methods

We have presented a wide variety of time-marching methods and shown howto derive their λ − σ relations. In the next chapter, we will see that thesemethods can have widely different properties with respect to stability. Thisleads to various trade-offs which must be considered in selecting a methodfor a specific application. Our presentation of the time-marching methods inthe context of a linear scalar equation obscures some of the issues involvedin implementing an implicit method for systems of equations and nonlinearequations. These are covered in this section.

6.10.1 Application to Systems of Equations

Consider first the numerical solution of our representative ODE

u′ = 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 (6.78)

Solving for un+1 gives

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106 6. TIME-MARCHING METHODS FOR ODE’S

un+1 =1

1− λh(un + heµh · aeµhn) (6.79)

This calculation does not seem particularly onerous in comparison with theapplication of an explicit method to this ODE, requiring only an additionaldivision.

Now let us apply the implicit Euler method to our generic system of equa-tions given by

~u′ = A~u− ~f(t) (6.80)

where ~u and ~f are vectors and we still assume that A is not a function of ~uor t. Now the equivalent to Eq. 6.78 is

(I − hA)~un+1 − ~un = −h~f(t + h) (6.81)

or

~un+1 = (I − hA)−1[~un − h~f(t + h)] (6.82)

The inverse is not actually performed, but rather we solve Eq. 6.81 as alinear system of equations. For our one-dimensional examples, the systemof 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 mul-tidimensions the bandwidth can be very large. In general, the cost per timestep of an implicit method is larger than that of an explicit method. The pri-mary 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

dt= F (u, t) (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 nonlinearODE

du

dt+

12u2 = 0 (6.85)

solved using implicit Euler time marching, giving

un+1 + h12u2

n+1 = un (6.86)

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6.10 Implementation of Implicit Methods 107

which requires a nontrivial method to solve for un+1. There are several differ-ent approaches one can take to solving this nonlinear difference equation. Aniterative method, such as Newton’s method (see below), can be used. In prac-tice, the “initial guess” for this nonlinear problem can be quite close to thesolution, since the “initial guess” is simply the solution at the previous timestep, 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 byconsidering Eq. 6.83. In order to implement the linearization, we expandF (u, t) about some reference point in time. Designate the reference value bytn and the corresponding value of the dependent variable by un. A Taylorseries expansion about these reference quantities gives

F (u, t) = F (un, tn) +(

∂F

∂u

)

n

(u− un) +(

∂F

∂t

)

n

(t− tn)

+12

(∂2F

∂u2

)

n

(u− un)2 +(

∂2F

∂u∂t

)

n

(u− un)(t− tn)

+12

(∂2F

∂t2

)

n

(t− tn)2 + · · · (6.87)

On the other hand, the expansion of u(t) in terms of the independent variablet is

u(t) = un + (t− tn)(

∂u

∂t

)

n

+12(t− tn)2

(∂2u

∂t2

)

n

+ · · · (6.88)

If t is within h of tn, both (t − tn)k and (u − un)k are O(hk), and Eq. 6.87can be written

F (u, t) = Fn +(

∂F

∂u

)

n

(u− un) +(

∂F

∂t

)

n

(t− tn) + O(h2) (6.89)

Notice that this is an expansion of the derivative of the function. Thus, rel-ative to the order of expansion of the function, it represents a second-order-accurate, locally-linear approximation to F (u, t) that is valid in the vicinityof the reference station tn and the corresponding un = u(tn). With this weobtain the locally (in the neighborhood of tn) time-linear representation ofEq. 6.83, namely

du

dt=

(∂F

∂u

)

n

u +(

Fn −(

∂F

∂u

)

n

un

)+

(∂F

∂t

)

n

(t− tn) + O(h2)

(6.90)

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108 6. TIME-MARCHING METHODS FOR ODE’S

Implementation of the Trapezoidal Method. As an example of howsuch an expansion can be used, consider the mechanics of applying the trape-zoidal method for the time integration of Eq. 6.83. The trapezoidal methodis given by

un+1 = un +12h[Fn+1 + Fn] + hO(h2) (6.91)

where we write hO(h2) to emphasize that the method is second order accu-rate. Using Eq. 6.89 to evaluate Fn+1 = F (un+1, tn+1), one finds

un+1 = un +12h

[Fn +

(∂F

∂u

)

n

(un+1 − un) + h

(∂F

∂t

)

n

+ O(h2) + Fn

]

+hO(h2) (6.92)

Note that the O(h2) term within the brackets (which is due to the local lin-earization) 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 up-dated at the end of each time step and the trapezoidal time march combineto make a second-order-accurate numerical integration process. There are,of course, other second-order implicit time-marching methods that can beused. The important point to be made here is that local linearization updatedat each time step has not reduced the order of accuracy of a second-ordertime-marching process.

A very useful reordering of the terms in Eq. 6.92 results in the expression[1− 1

2h

(∂F

∂u

)

n

]∆un = hFn +

12h2

(∂F

∂t

)

n

(6.93)

which is now in the delta form which will be formally introduced in Section12.6. In many fluid mechanics applications the nonlinear function F is not anexplicit function of t. In such cases the partial derivative of F (u) with respectto t is zero, and Eq. 6.93 simplifies to the second-order accurate expression

[1− 1

2h

(∂F

∂u

)

n

]∆un = hFn (6.94)

Notice that the RHS is extremely simple. It is the product of h and the RHSof the basic equation evaluated at the previous time step. In this example, thebasic 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 somefluid flow problem.

A numerical time-marching procedure using Eq. 6.94 is usually imple-mented as follows:

1. Solve for the elements of h~Fn, store them in an array say ~R, and save ~un.

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6.10 Implementation of Implicit Methods 109

2. Solve for the elements of the matrix multiplying ∆~un and store in someappropriate manner making use of sparseness or bandedness of the matrixif 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 valueof ~un and the process is repeated.

Implementation of the Implicit Euler Method. We have seen that thefirst-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 theexplicit dependence on time, we arrive at the form

[1− h

(∂F

∂u

)

n

]∆un = hFn (6.96)

We see that the only difference between the implementation of the trapezoidalmethod and the implicit Euler method is the factor of 1

2 in the brackets of theleft side of Eqs. 6.94 and 6.96. Omission of this factor degrades the methodin time accuracy by one order of h. We shall see later that this method is anexcellent choice for steady problems.

Newton’s Method. Consider the limit h → ∞ of Eq. 6.96 obtained bydividing both sides by h and setting 1/h = 0. There results

−(

∂F

∂u

)

n

∆un = Fn (6.97)

or

un+1 = un −[(

∂F

∂u

)

n

]−1

Fn (6.98)

This is the well-known Newton method for finding the roots of the nonlinearequation F (u) = 0. The fact that it has quadratic convergence is verifiedby a glance at Eqs. 6.87 and 6.88 (remember the dependence on t has beeneliminated for this case). By quadratic convergence, we mean that the errorafter a given iteration is proportional to the square of the error at the previ-ous iteration, where the error is the difference between the current solution

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110 6. TIME-MARCHING METHODS FOR ODE’S

and the converged solution. Quadratic convergence is thus a very powerfulproperty. 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 previousiteration. The reader should ponder the meaning of letting h → ∞ for thetrapezoidal method, given by Eq. 6.94.

6.10.4 Local Linearization for Coupled Sets of NonlinearEquations

In order to present this concept, let us consider an example involving somesimple boundary-layer equations. We choose the Falkner-Skan equations fromclassical boundary-layer theory. Our task is to apply the implicit trapezoidalmethod to the equations

d3f

dt3+ f

d2f

dt2+ β

(1−

(df

dt

)2)

= 0 (6.99)

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 =d2f

dt2, u2 =

df

dt, u3 = f (6.100)

This gives the coupled set of three nonlinear equations

u′1 = F1 = −u1 u3 − β(1− u2

2

)

u′2 = F2 = u1

u′3 = F3 = u2 (6.101)

and these can be represented in vector notation as

d~u

dt= ~F (~u) (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, ratherthan a simple nonlinear scalar function. The required extension requires theevaluation of a matrix, called the Jacobian matrix.9 Let us refer to this matrixas A. It is derived from Eq. 6.102 by the following process

A = (aij) = ∂Fi /∂uj (6.103)9 Recall that we derived the Jacobian matrices for the two-dimensional Euler equa-

tions in Section 2.2

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6.10 Implementation of Implicit Methods 111

For the general case involving a 3 by 3 matrix this is

A =

∂F1∂u1

∂F1∂u2

∂F1∂u3

∂F2∂u1

∂F2∂u2

∂F2∂u3

∂F3∂u1

∂F3∂u2

∂F3∂u3

(6.104)

The expansion of ~F (~u) about some reference state ~un can be expressed ina way similar to the scalar expansion given by eq 6.87. Omitting the explicitdependency on the independent variable t, and defining ~Fn as ~F (~un), onehas 10

~F (~u) = ~Fn + An

(~u− ~un

)+ O(h2) (6.105)

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

d~u

dt= An

~u +[~Fn −An

~un

]

︸ ︷︷ ︸“constant”

+O(h2) (6.106)

which is a locally-linear, second-order-accurate approximation to a set ofcoupled nonlinear ordinary differential equations that is valid for t ≤ tn + h.Any first- or second-order time-marching method, explicit or implicit, couldbe used to integrate the equations without loss in accuracy with respect toorder. The number of times, and the manner in which, the terms in theJacobian matrix are updated as the solution proceeds depends, of course, onthe 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

A =

− u3 2βu2 −u1

1 0 00 1 0

(6.107)

The student should be able to derive results for this example that are equiv-alent to those given for the scalar case in Eq. 6.93. Thus for the Falkner-Skanequations the trapezoidal method results in10 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 ofhigher order.

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112 6. TIME-MARCHING METHODS FOR ODE’S

1 + h2 (u3)n −βh(u2)n

h2 (u1)n

− h2 1 0

0 −h2 1

(∆u1)n

(∆u2)n

(∆u3)n

=h

− (u1u3)n − β

(1− u2

2

)n

(u1)n

(u2)n

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 iteratingwithin 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 givenby 1, 1, 2, 3, 5, 8, . . . Note that the series can be expressed as the solution toa difference equation of the form un+1 = un + un−1. What is u25? (Let thefirst term given above be u0.)

6.2 The trapezoidal method un+1 = un + 12h(u′n+1 +u′n) 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 =13

[4un − un−1 + 2hu′n+1

]

(a) Write the O∆E for the representative equation. Identify the polynomialsP (E) and Q(E).

(b) Derive the λ-σ relation. Solve for the σ-roots and identify them as prin-cipal or spurious.

(c) Find erλ and the first two nonvanishing terms in a Taylor series expansionof the spurious root.

6.4 Consider the time-marching scheme given by

un+1 = un−1 +2h

3(u′n+1 + u′n + u′n−1)

(a) Write the O∆E for the representative equation. Identify the polynomialsP (E) and Q(E).

(b) Derive the λ− σ relation.(c) Find erλ.

6.5 Find the difference equation which results from applying the Gazdagpredictor-corrector method (Eq. 6.65) to the representative equation. Findthe λ-σ relation.

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Exercises 113

6.6 Consider the following time-marching method:

un+1/3 = un + hu′n/3un+1/2 = un + hu′n+1/3/2

un+1 = un + hu′n+1/2

Find the difference equation which results from applying this method tothe representative equation. Find the λ-σ relation. Find the solution to thedifference equation, including the homogeneous and particular solutions. Finderλ and erµ. What order is the homogeneous solution? What order is theparticular solution? Find the particular solution if the forcing term is fixed.

6.7 Write a computer program to solve the one-dimensional linear convectionequation 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 theexplicit Euler and AB2 methods, use a Courant number, ah/∆x, of 0.1; for theother methods, use a Courant number of unity. Plot the solutions obtainedat t = 1 compared to the exact solution (which is identical to the initialcondition).

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 solu-tion at t = 1 using 2nd-order centered differences in space coupled with the4th-order Runge-Kutta method for grids of 100, 200, and 400 nodes. On alog-log scale, plot the error given by

√√√√M∑

j=1

(uj − uexactj )2

M

where M is the number of grid nodes and uexact is the exact solution. Findthe global order of accuracy from the plot.

6.10 Using the computer program written for problem 6.8, repeat problem6.9 using 4th-order (noncompact) differences in space.

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114 6. TIME-MARCHING METHODS FOR ODE’S

6.11 Write a computer program to solve the one-dimensional linear convec-tion equation with inflow-outflow boundary conditions and a = 1 on thedomain 0 ≤ x ≤ 1. Let u(0, t) = sin ωt with ω = 10π. Run until a pe-riodic steady state is reached which is independent of the initial conditionand plot your solution compared with the exact solution. Use 2nd-order cen-tered differences in space with a 1st-order backward difference at the outflowboundary (as in Eq. 3.69) together with 4th-order Runge-Kutta time march-ing. Use grids with 100, 200, and 400 nodes and plot the error vs. the numberof 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 differ-ences. Use a third-order forward-biased operator at the inflow boundary (asin Eq. 3.67). At the last grid node, derive and use a 3rd-order backward op-erator (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, andj + 1; see problem 3.1 in Chapter 3).

6.13 Using the approach described in Section 6.6.2, find the phase speederror, erp, and the amplitude error, era, for the combination of second-ordercentered differences and 1st, 2nd, 3rd, and 4th-order Runge-Kutta time-marching at a Courant number of unity. Also plot the phase speed errorobtained using exact integration in time, i.e., that obtained using the spatialdiscretization alone. Note that the required σ-roots for the various Runge-Kutta methods can be deduced from Eq. 6.69, without actually deriving themethods. Explain your results.