A0-stable linear multistep formulas of the-type

BIT 27 (1987), 123-128

Ao-STABLE LINEAR MULTISTEP F O R M U L A S OF THE s-TYPE GARY K. R O C K S W O L D

Computer Science Department, Mankato State University, Box 14, Mankato, Minnesota 56001, U.S.A.

Abstract. The or-type linear multistep formulas are a generalization of the Adams-type formulas. This paper is concerned with completely characterizing the Ao-stability of the k-step, order k ~-type formulas. Specifically, all such formulas of orders 4 or less are identified and it is shown that no ~t-type formulas of order 5 or more exist. These theorems generalize some previous results. AMS(MOS) subjective classification. 65L05.

Keywords: Ao-stability and multistep formulas.

In this paper our concern is the stability of linear multistep formulas (LMF) for the numerical solution of the ordinary differential equation

(1)

y'(x) = f ( x , y(x)),

x s [a, b]

y(a) = Yo where f is continuous and uniformly Lipschitzian with respect to the second argument. It is assumed that the interval I-a, b] is partitioned with a uniform step size h such that a = Xo < xl < x2 < ... < xm = b with mh = b - a . The k-step L M F for (1) is given by k

(2)

k

Z ~,Yn+, = h E fliY'n+,

i=0

i=0

where a~ and fl~ are constants. In addition, yj and y~ are numerical a p p r o x i m a tions to y(xj) and y'(xj) for j = 0, 1, 2 ..... m. An s-type formula is a k-step L M F with ct~ = 1, 0Ok_1 = --0C, ~k-2 = a - - I , ~ = 0 for i = 0, i, 2 ..... k - 3 and 0 < 0~ < 2. T h e s-type formulas are i m p o r t a n t for at least two reasons. First, they reduce to the well-known A d a m s - t y p e formulas when ~t = 1. Second, like the A d a m s - t y p e formulas, they have been

Received September 1985.

Revised November 1986.

124

G A R Y K. R O C K S W O L D

shown to be zero-stable in a variable coefficient implementation for all order and step size changes in [9] and more generally in [10]. Our stability study is limited to the test equation y' = )~y where 2 < 0. The characteristic polynomial associated with (2) on this test equation is Zk.~(~) = ~k(¢)+Vak(~),

v = ]hAl

where k

~k(~) =

k

Z ~#,

i=0

~,(~) =

Y~ B;~ '.

i=0

Formula (2) is Ao-stable for a fixed k when the roots of Xg, ~. have modulus less than 1 for all v > 0. In analyzing A0-stability the following transformation is helpful. l+z ~-1 ~(z) = 1 - z ,--, z(~) - 4 + 1 I--,7,

Sk(z)

k

=

k

(~(z)) -

~

s~z'

i=0 k

Xk, v(z) = Rk(z)+vSk(z) -- Z ti(v) zi" i=O

The mapping z sends {~ : ~ < 1} to {z : Re(z) < 0} and {~ : ~ = 1} to {z: Re(z) = 0}. The L M F (2) will be Ao-stable if and only if for all v > 0, Xk, v(z) is a Hurwitz polynomial, i.e., a polynomial whose roots all have a negative real part. In the determination of a Hurwitz polynomial the following result is used [1]. PROPOSITION. Let p(z) be a polynomial p(z) = ao + a j z +a2Y~"+a3z 3 + . . . of degree n @ 0 with real coefficients. Let (p(z)h be the "reduced" polynomial of degree n - 1 defined by (p(z)h = alal + (alaz -aoa3)z +ala3 z2 + (ala4-aoas)z 3 +...

At-STABLE L I N E A R MULTISTEP F O R M U L A S OF T H E c~-TYPE

125

Then, p(z) is a Hurwitz polynomial if and only if 1) aoai > for i = 1, 2 ..... n, and 2) (p(z)) 1 is a Hurwitz polynomial. An L M F is damped at infinity if all roots of ak(~) have modulus less than 1. It follows that (2) is damped at infinity if and only if S,(z) is a Hurwitz polynomial. In any k-step a-type formula of order k there are two free parameters among the coefficients. In this discussion ~ and flu have been chosen as these free parameters. It is the specific concern of this paper to completely characterize the A0-stability of k-step ~-type formulas of order k. Let O(k) denote the statement that "there exists an A0-stable k-step ~-type formula of order k." It is readily verifiable that O(k) is true for k = 1, 2 if and only if flk > 1/2. When flk > 1/2 the s-type formula is also damped at infinity. The following two theorems are direct results of the above discussion. Since the proofs are straightforward but algebraically cumbersome they have been omitted. However, the proofs to these theorems can be found in [5]. THEOREM 1. 0(3) is true if and only if the following two conditions both hold. 1) ½ < ~ < 2

2)

lO+a 16-c~ 2 < f13 < 24 2 4 ( 2 - ~)"

THEOREM 2. 0(4) is true if and only if the following two conditions both hold. 1) 1 < ~ < 2 2) 9_~x_< f14 <

a3 + ~2 _ 40c¢ +

128

2 4 ( 4 - ~)2

Using the definition found in [3] the error constant for the order 3 method is C3 = (8 + c~-24fl3)/(24(2-~)) and for the order 4 method is C4 = (232 + 1 9 ~ 720fl4)/(720(2-~)). As a result of this, it is not possible to make the above methods both order k + 1 and At-stable. The ~-type formulas will be both damped at infinity and A(0)-stable when the inequalities in Theorems Land 2 are replaced with strict inequality. (The A(0)-stability can be verified using the results from [4].) In Theorem 2, if • = 1 then this results in the unique A0-stable (but not A(0)-stable) Adams-type formula found in [6] and [7]. Since stability is often bought at the price of accuracy, it is of interest to determine the stability of a method versus its error constant. If one defines A (~)-stability as found in [8], henceforth called A (0)-stability (for obvious reasons), one will find that stability generally increases as ~ --, 2 and Ck --* ~ . However, Table 1 and Table 2 have been generated numerically to demonstrate that there

126

GARY K. ROCKSWOLD

exist methods of the a-type reasonable for practical use. In both tables, D represents the modulus of the largest root of ak(~), i.e., the damping at infinity. Table 1. Characteristics of the 3-step order 3 a-type formula.

.750 1.00 1,00 1.00 1.25 1.25 1.25 1.50 1,50 1.50 1.75 1.75 1.75 1.875 1.875 1.875

Table

2.

f13

C3

D

0 (degrees)

.498 .508 .558 .608 .5t9 ,619 .769 ,529 .729 .979 .540 .790 .990 .545 .745 .945

-.107 -.t33 -,183 -.233 -.178 -.311 -.444 -.267 -.667 - 1.167 -.533 - 1.533 -2.333 - 1.067 -2.667 -4,267

.968 ,813 ,914 .982 .652 .845 ,946 .530 .813 .95i .753 .758 .823 ,875 .876 .876

63.5 77.9 74.4 66.7 83,7 80.9 75.5 87.0 84.5 78.7 89.1 88,2 87.2 89.7 89.4 89,2

Characteristics of the 4-step order 4 c~-type formula.

t.25 1,25 t.25 1.50 1.50 1.50 1,75 1.75 1,75 1.875 1.875 1.875

f14

C4

D

0 (degrees)

.430 .440 .447 .440 .455 .463 .450 .458 ,470 .456 .466 .478

- .099 - 1.125 - 1.225 -.156 -.186 -.201 -.328 -.358 - .408 -.671 - .751 -.851

.899 .954 .99t .796 .872 .904 .907 .751 .804 .924 ,874 .875

50,0 47.7 40.2 63,6 63.7 62.5 71.4 72.6 72.7 74.3 75.6 75.8

It remains to be shown that O(k) is false whenever k > 5. It is known [3] that formula (2) is order p if and only if

Sk(z) -- Rg(z)C(z) + O(zp ) 2

(3)

where g

c(z) =

In ~(z)

= ½-

+

6 +... +

+...],

127

A0-STABLE LINEAR MULTISTEP FORMULAS OF THE or-TYPE

with c2~ > 0, i >_ 1. For any a-type formula we have 1-z k l+z k Rk(z) = ( ~ - ) [ ( 1 - ~ - - z ) - - e ( l + z ) k - \ ~ - -] z A

Using (3) above it can be shown that

Sk(z)

=

+ (terms of degree 1 to k - 2) + ~-~(2 - e) for k > 5.

(Itisassumedthat(~)=Oifn 0 and si < 0, 1 _< i, j < k, where Sk(z) = ~ = osf. Then for sufficiently large v > 0, there must exist ti(v)> 0 and t~(v)< 0, where Xk.~(Z)= ~,~=oti(v)Zi. Thus. X k , v(Z) is not a Hurwitz polynomial by the proposition and O(k) is false. II THEOREM 3. O(k ) is false for k >- 5. PROOF. The proof will rely on Lemma 4 by noting that the coefficient of

z k- 1 in Sk(Z) is negative whenever k > 5, whereas the constant term in S~(z) is always positive. For k = 5 the

Zk - 1

coefficient is 1[49 98] 241_90 ~ - 9-0

which is negative whenever 0 < ~ < 2. For k > 6 the coefficient of z k- 1 is less than (4)

1

1 [ - / k - 2 ) ( k~t+ -2) k-3

(E-c0 1 = 1~ [ _ ~ k 2 + ( 7 ~ _ 4 ) k _ 4 ~ + 8 ] "

128

Ao-STABLELINEAR MULTISTEP FORMULASOF THE ~-TYPE

The derivative ~ 2 ( - 2 e k + 7 e - 4 ) is < 0 if k >_ 6 a n d hence expression (4) is m a x i m u m when k = 6. Thus, the z k-1 coefficient is less t h a n (c~-8)/6 which is negative for 0 < e < 2. [] T h e a b o v e t h e o r e m s generalize the results found in [2]. In [5] the e - t y p e f o r m u l a s are investigated in a variable coefficient i m p l e m e n t a t i o n for the solution of stiff differential equations.

Acknowledgements. T h e a u t h o r w o u l d like to t h a n k Professor R. J. L a m b e r t a n d the referees for their v a l u a b l e suggestions.

REFERENCES 1. R. J. Duffin, Algorithms for classical stability problems, SIAM Rev., 11 (1969), 196-213. 2. R. B. Feinberg, Ao-stableformulas of Adams-type, SIAM J. Numer. Anal., 19 (1982), 259-262. 3. P. Henfici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley, New York, 1962. 4. R. Jeltsch, Stiff stability and its relation to A o- and A(O)-stability. SIAM J. Numer. Anal., 13 (1976), 8-17. 5. G. K. Rockswold, Stable variable step stiff methods for ordinary differential equations, Doctoral thesis, Department of Mathematics, Iowa State University, 1983. 6. D. J. Rodabaugh and S. Thompson, Low-order Ao-stable Adams-type correctors, J. Comput. Appl. Math., 5 (1979), 225-233. 7. M.J. Strassberger, Families of stiffly stable Adams type linear multistep formulas, Doctoral thesis, Department of Mathematics, Iowa State University, 1980. 8. O. B. Widlund, A note on unconditionally stable linear multistep methods, BIT, 7 (1967), 65-70. 9. Z. Zlatev, Stability properties of variable step-size variable formula methods, Numer. Math., 37 (1978), 175-182. 10. Z. Zlatev, Zero-stability properties of the three-ordinate variable step-size variable formula methods, Numer. Math. 37 (1981), 157-166.