Numerical Methods for Ordinary Dierential Equations Episode 7 potx

The alternative example of a method of this order that we give uses C2 and D2 with subsidiary conditions to repair the gaps in the order conditions caused by C2 not applying to stage 2 a

We can eliminate the factor c6−1 because, if it were zero, then it would follow

that c3= 13 and that c4 = 1, which are consistent with the vanishing of thesecond factor, which leads to

Having chosen c3, and therefore c4, together with arbitrary c2, c5 and c6 and

the known value c7 = 1, excluding some impossible cases, we can solve for

the components of b from (326a)–(326g) We can then solve for a54, a64 and

a65 from the consistent equations (326h)–(326k) We then solve for a32 from

(326l) and then for a42, a43, a52, a53, a62 and a63 from (326l) with i = 4, 5, 6

and from (326o), (326p) and (326q) It remains to compute the first column

of A from (326m) and the last row from (326n).

The following example is of a method derived from these equations:

0

1

3

1 3 2

1

3

1 12

1

3 −1

12 5

15 8 1

1 −261

260

33 13

43

156 −118

39 32 195 80 39 13

200 0 1140 1140 254 254 20013

It is possible to derive sixth order methods in other ways For example,

Huˇta used the C(3) with subsidiary conditions for stages 2 and 3 However,

he used s = 8, and this gave him more freedom in the choice of c.

The alternative example of a method of this order that we give uses C(2) and D(2) with subsidiary conditions to repair the gaps in the order conditions caused by C(2) not applying to stage 2 and D(2) not holding for stage 6 It

is necessary to choose b2= 0, and to require that c3, c4 and c5 are related sothat the right-hand side vanishes in the equations

because the left-hand sides are identically zero A method derived along theselines is as follows:

0

2

5

2 5 4

5 2

15 −44

675 −88

135

76 351

336 325

689 −324

689

45 106

1 −2517

4864 −55

38

10615 31616

567 7904

7245 4864 2597 2432

4992

6561 20384

3375 12544

53 768 19 294

327 Methods of orders greater than 6

Methods with order 7 must have at least nine stages It is possible to constructsuch a method using the principles of Subsection 323, extending the approachused in Subsection 326 The abscissa vector is chosen as

c = [ 0 1

3c4 2

3c4 c4 c5 c6 c7 0 1 ] , and the orders of stages numbered 4, 5, , 9 are forced to be 3 To achieve

consistency of the conditions

3− 12u + 24v + 14u2− 70uv + 105v2,

where u = c4+ c5 and v = c4c5 The value of c7 is selected to ensure that

 1

x(1 − x)(x − c4)(x − c5)(x − c6)(x − c7)dx = 0.

The tableau for a possible method derived along these lines is

11

148

1331 0 1331150 − 56

1331 2

3 −404

243 0 −170

27

4024 1701

10648 1701 6

0 1545 0 0 53996 −1815

20384 −405

2464

49 1144

1 −113

32 0 −195

22

32 7

29403

3584 −729

512 1029 1408

21 16

Exercises 32

32.1 Find a method with s = p = 3 such that c = [0,12, 1].

32.2 Find a method with s = p = 3 such that c = [0,1

3, 1].

32.3 Find a method with s = p = 4 such that b1= 0 and c2= 15

32.4 Find a method with s = p = 4 such that b2= 0 and c2= 14

32.5 Find a method with s = p = 4 such that b1= 0 and c3= 0

32.6 Show that Lemma 322A can be used to prove that c4= 1, if s = p ≥ 4.

32.7 Show that Lemma 322A can be used to prove that c5= 1, if s = p ≥ 5

leading to an alternative proof of Theorem 324B

33 Runge–Kutta Methods with Error Estimates

330 Introduction

Practical computations with Runge–Kutta methods usually require a means

of local error estimation This is because stepsizes are easy to adjust so as

to follow the behaviour of the solution, but the optimal sequence of stepsizesdepends on the local truncation error Of course, the exact truncation errorcannot realistically be found, but asymptotically correct approximations to

it can be computed as the integration proceeds One way of looking at this

is that two separate approximations to the solution at a step value x n arefound Assuming that the solution value at the previous point is regarded

as exact, because it is the local error that is being approximated, denote the two solutions found at the current point by yn and yn Suppose the two approximations have orders p and q, respectively, so that

y n = y(xn) + O(h p+1 ), yn = y(xn) + O(h q+1 ).

Then, if q > p,

yn − y n = y(xn) − y n + O(h p+2 ),

which can be used as an approximation to the error committed in the step

Furthermore, the approximation becomes increasingly accurate as h becomes

small Thus yn − y n is used as the error estimator

Even though we emphasize the construction of method pairs for which

q = p+1, and for which it is y n(rather than the asymptotically more accurateapproximationyn ) that is propagated as the numerical approximation at x n,

customary practice is to use the higher order as the propagated value This

is sometimes interpreted as ‘local extrapolation’, in the sense that the errorestimate is added to the approximate solution as a correction While theestimator is still used as a stepsize controller, it is now no longer relatedasymptotically to the local truncation error

We review the ‘deferred approach to the limit’ of Richardson (1927) andthen consider specially constructed Runge–Kutta tableaux, which combinetwo methods, with orders one apart, built into one The classical method

of this type is due to Merson (1957), but we also consider built-in estimatorsdue to Fehlberg (1968, 1969), Verner (1978) and Dormand and Prince (1980).Some of the methods derived for the author’s previous book (Butcher, 1987)will also be recalled

331 Richardson error estimates

Richardson extrapolation consists of calculating a result in a manner thatdepends on a small parameter, and for which the error in the calculationvaries systematically as the parameter varies By using a sequence of values

of the parameter, much of the effect of the errors can be eliminated so that

improved accuracy results In numerical quadrature, for example, the method

of Romberg (1955) is based on calculating an integral I = b

a φ(x)dx using

the trapezoidal rule with a stepsize h equal to an integer divisor of b − a For

a single choice of h, the result computed can be expanded by an asymptotic

formula of the form

we obtain an ‘improved’ sequence in which the C1H2 terms are eliminatedfrom the asymptotic expansions so that convergence towards the exact result

I is more rapid as terms in the sequence are calculated Similarly, a second

sequence of improved approximations can be found from

c2 a21

c3 a31 a32

. . .

c s a s1 a s2 · · · a s,s −1

b1 b2 · · · b s −1 b s

Repeating the calculation with h replaced by 12h but carrying out two steps,

rather than only one, is equivalent to taking a single step with the original h,

but using the tableau

y n = y(xn)− C(x n)h p+1 + O(h p+2 ), (331a)then

yn = y(x n)− 2 −p C(x

n )h p+1 + O(h p+2 ), (331b)because the error in computing y n is 2−p−1 C(x n)h p+1 + O(h p+2) contributedfrom each of two steps

From the difference of (331a) and (331b) we find

yn − y n = (1− 2 −p )C(xn)h p+1 + O(h p+2 ),

so that the local truncation error in yn can be approximated by

(1− 2 −p)−1(y − y n). (331c)

This seems like an expensive way of computing the error in the result

computed using an s-stage method, because the additional computations

required for the estimation take twice as long as the result itself However, the

additional cost becomes more reasonable when we realize that it is not ynbut

yn that should be propagated The additional cost on this basis is somethinglike 50% Actually, it is slightly less than this because the calculation of the

derivative of y n −1 is shared by each of the two methods, and needs to be

carried out only once

332 Methods with built-in estimates

Instead of using the Richardson technique it is possible to combine twomethods into one by constructing a tableau with common stages but twoalternative output coefficient vectors The following method, due to Merson(1957), seems to have been the first attempt at constructing this type ofstepsize control mechanism:

01 3 1 3 1 3 1 6 1 6 1

2 1

10 0 103 25 15

The interpretation of this tableau, which contains two b vectors, is that it

combines two methods given by

01 3 1 3 1 3 1 6 1 6 1 2 1

2 1

with eccentricities e = 0.1, e = 0.5 and e = 0.9, respectively.

333 A class of error-estimating methods

In the search for efficient step-control mechanisms, we consider (s + 1)-stage

methods of the form

in the succeeding step It is convenient to write order conditions for the

embedded method pair in terms of the number B = bs+1 and the artificialtableau

B by the derivative of the order p result found by the method represented by

(333b) This enables us to form modified order conditions for (333c), whichwill ensure that both (333a) and (333b) satisfy the correct conditions We

denote the elementary weights for (333c) by Φ(t).

Theorem 333A If (333b) has order p and (333a) has order p + 1 and

Proof For a given tree t, let  Φ(t) denote the elementary weight for (333a) and Φ(t) the elementary weight for (333b) Because the latter method has order

p, it follows that for a tree t = [t1t2· · · t m], with order not exceeding p + 1, we have Φ(ti) = 1/γ(ti), for i = 1, 2, , m Hence, for a method identical with (333a) except for b replaced by the basis vector e s+1, the elementary weight

Adding B multiplied by this quantity to Φ(t) gives the result

To prove the converse, we first note that, because B = 0, the previous

argument can be reversed That is, if (333b) has order p then (333d) implies that (333a) has order p + 1 Hence, it is only necessary to prove that (333b) has order p We calculate Φ(t), for r(t) ≤ p as follows, where we have written

χ i (t) for the coefficient of b i in Φ(t)

is the vector of coefficients in the proposed error estimator That is,

it is the approximation to be propagated and, of course, the dashed line below

the b vector separates the order p+1 approximation from the error estimator.

Now let us look at some example of these embedded methods Methods oforders 1 and 2 are easy to derive and examples of each of these are as follows:

0

1 2 1 2

−1 2 1 2

and

01 2 1 2 1

2 0 1 2

1 6 1 3 1 3 1 6 1

6 1

3 −2 3 1 6

Observe that for the second order method, the third order method in which

it is embedded is actually the classical fourth order method

Order 3 embedded in order 4 requires s = 4 stages From the modified order

so that, equating the products (333h)×(333k) and (333i)×(333j) and

simplifying, we find the consistency condition

c4=1− 7B + 12B2

1− 6B + 12B2.

For example, choosing B = 121 to give c4 = 67, together with c2 = 27 and

c3=4, yields the tableau

02 7 2 7 4

7 −8 35

4 5 6

7 29

42 −2

3 5 6

11 96

7 24 35 96

7 48 1 12

−5 96

1

8 −5

96 −5

48 1 12

Order 4 embedded in order 5 requires s = 6 That is, there are seven stages

overall, but the last stage derivative is identical to the first stage derivativefor the following step To derive a method of this type, make the simplifyingassumption

The left-hand sides of (333m)–(333p) consist of only a single term and wesee that the product of (333m) and (333p) is equal to the product of (333n)

and (333o) Thus we obtain consistency conditions for the values of a65 and

a54 by comparing the products of the corresponding right-hand sides Afterconsiderable manipulation and simplification, we find that this consistencycondition reduces to

c6= 1− q0B

q0− q1B + q2B2, (333q)with

q0= 10c23c4+ 2c4− 8c3c4− c3,

q1= 60c2c4− 56c3c4+ 16c4− 8c3,

q2= 120c23c4− 120c3c4+ 40c4− 20c3.

Construction of the method consists of selecting c2, c3, c4, c5and B; choosing

c6 in accordance with (333q); evaluating a65 and a54 from the consistent

equations (333n), (333o) and (333p); and then evaluating a64 from (333l).The remaining coefficients are then evaluated using the remaining conditionsthat have been stated

An example of a method in this family is

13 50 4

5

548

7475

688 2875

572

2875 −88

575

132 299

165

16 135

50 351

575 2376

1 15

For p = 5, that is, a fifth order method embedded within a sixth order method, s = 8 seems to be necessary We present a single example of a method

satisfying these requirements For all stages except the second, the stage order

is at least 2, and for stages after the third, the stage order is at least 3 Underthese assumptions, together with subsidiary conditions, it is found that for

consistency, a relation between c4, c5, c6, c8 and B must hold Given that

these are satisfied, the derivation is straightforward but lengthy and will not

be presented here The example of a method pair constructed in this way isshown in the following tableau:

4 −93

22 0 249211408 −10059

704

735 1408

735 704 17

27949088

81711267−452648800

245133801

270189568 467982711

2 39

334 The methods of Fehlberg

Early attempts to incorporate error estimators into Runge–Kutta methods areexemplified by the work of Fehlberg (1968, 1969) In writing the coefficients

of methods from this paper, a tabular form is used as follows:

is a Runge–Kutta method of order p + 1 The additional vector d = ˆ b − b is

used for error estimation The fifth order method, with additional sixth orderoutput for error estimation, recommended by Fehlberg, is

1

6

1 6 4

15

4 75

16 75 2

3

5

6 −8

3 5 2 4

128 −11

80 55 128

5

66 0 07

1408 0 11252816 329 125768 0 665 665

−5

66 5 66 5 66

We also present a similar method with p = 7 This also comes from

Fehlberg’s paper, subject to the correction of some minor misprints Theaugmented tableau is

6

31

300 0 0 0 22561 −2

9 13 900 2

3 2 0 0 −53

6 704

45 −107 9 67

90 31

3 −91

108 0 0 10823 −976

135 311

54 −19 60 17

6 −1 12

1 23834100 0 0 −341

164 4496

1025 −301

82 2133 4100 45 82 45 164 18 41

205 0 0 0 0 −6

41 − 3

205 −3 41 3 41 6

41 0

1−1777

4100 0 0 −341

164 4496

1025 −289

82 2193 4100 51 82 33 164 12

41 0 141

105

9 35 9 35 9 280 9 280 41

The two methods presented here, along with some of the other Runge–Kutta pairs derived by Fehlberg, have been criticized for a reason associatedwith computational robustness This is that the two quadrature formulae

characterized by the vectors b and ˆ b are identical Hence, if the differential

equation being solved is approximately equal to a pure quadrature problem,then error estimates will be too optimistic

Although the methods were intended by Fehlberg to be used as order p

schemes together with asymptotically correct error estimators, such methodsare commonly implemented in a slightly different way Many numericalanalysts argue that it is wasteful to propagate a low order approximationwhen a higher order approximation is available This means that the method

(A, ˆ b , c ), rather than (A, b , c), would be used to produce output values The order p + 1 method will have a different stability region than that of the order

p method, and this needs to be taken into account Also there is no longer an

asymptotically correct error estimator available Many practical codes have no

trouble using the difference of the order p and order p + 1 approximations to

control stepsize, even though it is the higher order result that is propagated

335 The methods of Verner

The methods of Verner overcome the fault inherent in many of the Fehlbergmethods, that the two embedded methods both have the same underlyingquadrature formula The following method from Verner (1978) consists of afifth order method which uses just the first six stages together with a sixthorder method based on all of the eight stages Denote the two output coefficient

vectors by b and  b , respectively As usual we give the difference  b − b which

is used for error estimation purposes:

9 −2

81

4 27

8 81 2

1 −369

73

72 73

5380

219 −12285

584

2695 1752 8

9 −8716

891

656 297

80 0 254 1120243 16077 70073 0 057

65

1377 2240

121

320 0 8320891 35233

As for the Fehlberg methods, we have a choice as to whether we use thefifth or sixth order approximation as output for propagation purposes Eventhough the sixth order choice leaves us without an asymptotically correctlocal error estimator, the use of this more accurate approximation has definiteadvantages In Figure 335(i) the stability regions for the two approximationsare plotted It is clear that stability considerations favour the higher ordermethod