A-stable block implicit one-step methods

BIT 12 (1972), 252-266

A-STABLE

BLOCK

IMPLICIT

ONE-STEP

METHODS

H. A. WATTS and L. F. SHAMPINE Abstract. A class of methods for solving the initial value problem for ordinary differential equations is studied. We develop r-block implicit one-step methods which compute a block of r new values simultaneously with each step of application. These methods are examined for the property of A-stability. A sub-class of formulas is derived which is related to Newton-Cotes quadrature and it is shown that for block sizes r = 1, 2. . . . . 8 these methods are A-stable while those for r = 9,10 are not. We construct A-stable formulas having arbitrarily high orders of accuracy, even stiffly (strongly) A-stable formulas.

1. I n t r o d u c t i o n .

W e shall s t u d y a class of m e t h o d s for solving numerically the initial value problem for o r d i n a r y differential equations. These procedures, described as r-block implicit one-step methods, a d v a n c e the numerical solution b y a block of r new solution values a t a time. Ordinarily n u m erical schemes of R u n g e - K u t t a t y p e proceed a step at a time while performing several function evaluations within the step. Often one can i n t e r p r e t these internal c o m p u t a t i o n s as representing a p p r o x i m a t e values of the solution a t points lying within the step t h o u g h these values are generally m u c h less a c c u r a t e t h a n t h e solution value a t the e n d of the step. W i t h this in mind, it is n a t u r a l to e x t e n d the i n t e r p r e t a t i o n t o (implicit) processes which yield r acceptable values of the discrete solution within a step or block. F o r t h e practical use of these i n t e r m e d i a t e values it is desirable to h a v e the mesh-points of the block more or less e v e n l y spaced a n d to h a v e all values equally accurate. T h e block procedures m a y be considered t o be one-step m e t h o d s for theoretical purposes since only the last m e m b e r in a block will e n t e r into t h e formulas for c o m p u t i n g the n e x t block in the sequence. A n u m b e r of a u t h o r s h a v e studied implicit one-step methods, e . g , Ceschino a n d K u n t z m a n n [7], B u t c h e r [4, 5], Ehle [11] a n d Axelsson [2]. However, the n o t i o n of obtaining a block of values simultaneously w i t h each stage of application r a t h e r t h a n a single v a l u e over one step has n o t Received December 13, 1971. Revised February 11, 1972.

A-STABLE

BLOCK

IMPLICIT

OR~E-STEP METHODS

253

received as much attention. Indeed, the earliest references known to the authors--Milne [13], Collatz [8]--place emphasis on using such a method only to get starting values required for predictor-corrector schemes. A different approach based on global integration was presented b y Axelsson [1]. The procedure due to Clippinger and Dimsdale (cf. [16]) does represent a 2-block implicit scheme which they used for advancing the numerical solution. Cda [6] and Rosser [15] also present a detailed study of block implicit methods with the explicit intent of applying the algorithm repetitively for obtaining the numerical solution over the entire interval. In this paper we continue the study of general r-block implicit methods initiated in Shampine and Watts [16], with particular emphasis being placed on the property of A-stability. In section 2 we discuss the basic characteristics of block impliet methods while section 3 is devoted to the concept of A-stability as applied to these methods. In section 4 we examine a class of formulas which we shall call interpotatory formulas of Newton-Cotes type. We conclusively establish that the block methods for sizes r = 1 , 2 , . . . , 8 are A-stable whereas those for r = 9,10 are not. In section 5, using diagonal and sub-diagonal Padd approximations to the exponential function, we are able to construct A-stable formulas having arbitrarily high orders of accuracy. The formulas based on the sub-diagonal approximations are even "stiffly" (strongly) A-stable, an additional property valuable for numerical purposes.

2. Block implicit methods. We shall be interested in obtaining a numerical solution of (1)

u'(x) - - f ( x , u ( x ) ) ,

u(a) -- ~,

a < x < b,

where we make the usual assumptions that f is continuous and satisfies a Lipschitz condition on the region [a,b] × ( - 0% oo). Let us introduce the points x k = a + kh for some h e (0, ho). We wish to generate a sequence Yk approximating u(xk). B y an r-block method we shall mean a procedure which yields r additional values Yn+I,Y~,+~ . . . . , Yn+r simultaneously at each stage of application; here n = m r and m = 0 , 1 , . . . with X n + r = a + ( m + l ) r h < b . The formulas we shall study m a y be put in the form (2) Ym = ~Y~ + hdf~ + hBF(~m),

fi-f(xj,yt), B--~(bil), ~ - - ( 1 , 1 , . . . , 1 ) T, d - - ( d i , d 2. . . . . dr) T, ym = (Yn+l . . . . . y~+r) r, F ( f l m ) = ( f n + l . . . . . fn+r) T, and the initial value y o = ~ .

where

:BIT 12 - - 17

254

H . A. W A T T S A N D L . F . S t t A M P I N E

Equation (2) represents a system of non-linear equations for the new values which can be shown to have a unique solution if h is suitably restricted by using the contracting mapping principle (cf. [16]). This condition is typical but conflicts with the use of large h allowed by A-stable methods. In practice we m a y have to presume the existence of a solution and obtain it in a different way, e.g., Newton's method. If we define the local truncation error ~m of the procedure b y ~m = ~u(x~) + hdf(x,, u(x,)) + hBF(~m) + % , we can state a convergence theorem. THEOREM 1. Suppose we have an r-block implicit one-step method defined by (2), and let us assume the existence of p and 0 < q < p such that the truncation errors satisfy H~mll = O( hq+l ) and ]V~+rl = O(hP +l ). Then the method is convergent with global error of the order of h ~ where v = m i n ( p , q + 1). That is, H~m-~mH=O(h~) for each m = 0 , 1 . . . . such that x(m+l)r< b, and the method is said to be of order ~,. The proof can be found in [16]. A couple of remarks seem noteworthy: First, the proof regards the block procedure as an implicit one-step method with step size rh for calculating Yn,Y,~+r,"" Second, advantage is taken of formulas which are more accurate at the end of the block t h a n in the interior.

3. A-stability.

Here we make some general observations about block implicit methods and the property of A-stability [10]. Consider the scalar equation (3)

u'(x) = au(x),

u(O) = 1,

with (complex) constant a. A numerical method for the solution of (3) with Re~ < 0 is said to be A-stable if for Mt fixed but arbitrary h > 0 the resulting approximate solution y~ tends to zero as k -+ c~. Upon applying the block scheme (2) to the test equation (3), we find (4)

( I - 2B)y m = (~ + 2d)y,~ ,

where 2 =~h. The procedure must be applicable for all h > 0 and all with Re~ < 0 so it is necessary t h a t I - 2 B be invertible for all 2 with Re2 < 0. Let us define P(2), Pk(2) and a i, akt by (5)

P(2) = d e t ( I - 2 B ) = ~ ai2i , i=0

A-STABLE

(6)

BLOCK IMPLICIT

ONE-STEP METHODS

Pk(2) = det[Bk(2)] = ~ aki2 ~, i=0

255

k= l,2 . . . . . r ,

where Bk(2) is the matrix I - 2 B with the kth column replaced by the vector ~ + / d . The determinant definitions show t h a t P(2), Pk(2) are polynomials with degree at most r and normalized such t h a t P ( 0 ) = Pk(0) = 1. The necessary requirement on B is t h a t all roots of P(~)= 0, or what is the same, all eigenvahies of B must have non-negative real parts. I n what follows we suppose this condition is always satisfied. Using the above definitions and Cramer's rule in (4) we find t h a t (7)

Pk(2) Yn+k = p().) Yn,

=

LP~J

k=l,2,...,r,

LP(A) J "

F r o m this it is clear t h a t Ym -~ ~ as m -~ co if and only if IPr(2)/P(2)I < 1. This establishes THEOREM 2. A necessary and sufficient condition for an r-block method to be A-stable is that JPr(2)/P(2)I < 1 for all 2 having R e / < 0.

Hence, given an r-block method satisfying the necessary conditions, the study of A-stability is equivalent to the examination of the rational function PffP. The following lemma from Birkhoff and Varga [3] will suffice for m a n y of our applications. LEMMA 1. Suppose P(z) and Q(z) are real polynomials such that Q(z)= P ( - z) and the zeros of P(z) have positive real roots; that is P ( - z ) is a stable polynomial. Then JQ(z)/P(z)] < 1 for all Rez < 0. There are evidently two approaches to A-stability in this context. I n the next section we go from formulas studied by us and others to this rational function and so study their stability. In the remaining sections we show t h a t we can start with polynomials Pr*(2), P*(2) having the property t h a t tPr*(2)/P*(2)] < 1 for all 2 with Re2 < 0 and deduce a block method for which P(2)~-P*(2), Pr(2)--Pr*(2), t h a t is, an A-stable method. From this approach we can glean other useful properties of the method so constructed.

4. "Newton-Cotes" block implicit methods. In this section we shall discuss a class of formulas which are based upon quadrature formulas of the Newton-Cotes type. These are the formulas presented in Rosser [15] and also studied b y C6a [6]. Since the

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t I . A. W A T T S A N D L. F. S H A M P I N E

initial value problem u ' ( x ) = f ( x , u ( x ) ) , u ( x n ) = y ~, is equivalent to the integral equation u(x) = y~ + f~nf(t, u(t)) dr, we approximate the integrals in Xn+i 1o

u(x~+~) = y ~ + l f ( t , u ( t ) ) d t ,

j=1,2 ..... r,

xn

by integrating the rth degree interpolating polynomial which agrees with u'(x) at Xn,Xn+ 1. . . . ,x,~+r. Thus we obtain formulas of the form (2) with dj ~ aio, b~k =- ajk where 1 x,~(j alk = ~ ~ Ik(t)dt' ~n

j = 1,2,

""

. ,r

k=O, 1. . . . ,r

and the Ik(t ) are the fundamental Lagrangian polynomials. In [17], it is shown that the discretization errors of these formulas are given by J hr+~ i s(s-1)...(s-r)ds, (8) vt (r+l)!Ur+~(~j) j=l,2 .... ,r, 0

except that for even r, the stronger result obtains, (9)

h~+3 ur+3(~r) i s 2 ( s - 1 ) . . . ( s - r ) d s , rr -- (r+2)-----~. 0

where x~ < ~j < xn+ ~. The reader will recognize that the formula for vr is the quadrature error associated with the well-known Newton-Cotes r + 1 point formula. Asymptotic formulas for the error of the approximate solution obtained using this class of block methods are also presented in [17]. We shall refer to this class of formulas as "interpolatory formulas of Newton-Cotes type". Based on their derivation it is easy to see tha$ such formulas lead to an equivalent polynomial collocation method. I n fact, the extension of the discrete solution defined as the polynomial ~(x) (considering the y~+~ as given) ~(x) = y~ + ~ f ( x ~ + k , y ~ + k ) k=o

Ik(t)dt , X~

collocates at x n, Xn+~,. . . , Xn+ r. 4.1. M e t h o d of undetermined coefficients.

An equivalent way of defining interpolatory quadrature formulas using s + 1 nodes is to specify t h a t the quadrature formulas have degree

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ONE-STEP

METHODS

257

of precision at least s. Since we shall have later use of these results, let us now consider the method as applied to constructing the NewtonCotes interpolatory block formulas. Accordingly, we require t h a t the formulas (2) be exactly satisfied for y(x)=xV for all l _ _ < p < r + l (we have already forced them to satisfy the constraint for p = 0). We m a y assume t h a t h = 1 and our evenly spaced nodes x~ are 0 , 1 , . . . , r . Thus the b¢k and d¢ must satisfy the following equations r

di+~bik=j, k=l

(10) (11)

j=l,...,r

/-,~ bik ~ - 1 ---- ljp,

j

k=l

p = 2. . . . , r + 1

P

=1 ....

,r

Written in matrix form this is VT~i=~t where ~t=(d¢,bjl . . . . . b¢r)T, ~i=(j,j~]2 . . . . ,jr+l/(r+ 1)) r and Vr is the transpose of a Vandermonde matrix. We see t h a t by requiring the formulas (2) to have the highest algebraic degree of precision, the coefficients are determined as the unique solutions of (10), (11). 4.2. Properties of the Newton-Cotes formulas. The only thing t h a t we shall need to know about the Newton-Cotes formulas for our stability analysis is t h a t the corresponding r-block method has local accuracy O(hr+u). Because of this accuracy requirement, the solution of (3) satisfies (12)

Pk(~)

Yn+k = p(~) Yn = Ynek~+O(]~]r+2),

k=l,2,...,r

where the P(2), Pk(~) are defined in (5) and (6). The following lemmas are readily obtained from these relations. Complete details can be found

in [17]. LEMMA 2. For all r, Pk(2)=P(2)e~X+O(12] r+~) for each of k = 1,2 . . . . ,r and all ~ such that P(2) 4=0. LEMMA 3. The coefficients Ski of the polynomials Pk(2) are given by

k i-i j=o(i-j)! LE~t

i = O,l,...,r k = 1,2 . . . . . r .

4. The coefficients of the polynomial P(~) satisfy the equations k i-1

(13)

a*-i+l i! i=l

k~ (r+l)!'

k=l,2,...,r.

258

H, A. WATTS AND L . F . SHAMPINE

LENIMA 5. The coefficients of the polynomial P(2) are given by (14)

at =

(r-i+l)! (r+l)!

/*r-i,

i=0,1 ..... r

where the/xl are defined by

(x- 1)(x- 2)... (x-r) = ~ ~ f . 3"=0

Letting

(r+l)! fli =

j !

ar-~+1,

we recognize that (13) says that the fll . . . . . fir are the coefficients of the unique interpolating polynomial q(x)=~,flix ~-1 satisfying q(l~)=Ic r for k = 1,2 . . . . r. Noting that q(x) = x ~ - ( x - 1 ) ( x - 2 ) . . . ( x - r ) is the polynomial of interest, the result follows. LEMMA 6. For all r, Pk(~)=Pr-k(--~) for each of It= 0,1 . . . . . It/2], where we define Po(2)= P(2). After establishing that Pr(2)= P ( - 2), the remaining identities follow easily from the identities of Lemma 2. But this result can be shown b y using Lemma 2 to write Pk(2)=Pr(--2)e(r-k)~+O(]21r+u). In the same manner that the coefficients a i were determined, the coefficients ( - 1)iart of the polynomial P r ( - 4 ) are seen to satisfy exactly the same set of equations as did the a t. The result is concluded b y appealing to the uniqueness of solutions to Vandermonde matrix systems. From Lemmas 1,6 and Theorem 2 we conclude that the r-block procedure defined b y the Newton-Cotes type formulas will be A-stable if the zeros of P(2) have positive real parts, i.e., if P ( - ~ ) is a stable polynomial or - B is a stable matrix. Since (14) gives explicit formulas for the coefficients ( - 1 ) i a t for P ( - ~), we applied the g o u t h algorithm [9] numerically to establish LEMMA 7. All zeros of P(2) have positive real parts for the cases r = 1,2 . . . . . 8. For r = 9, 10 there exists a zero of P(~) having negative real part. We remark that Wright [18] has also established the A-stability properties of methods equivalent to these from the viewpoint of collocation. He also used the R o u t h algorithm numerically to obtain results similar to Lemma 7. Because of roundoff errors in floating point computation, his computation was not conclusive. Since the coefficients of the polynomials are rational and the R o u t h scheme involves only rational arithmetic, it is possible (for a limited number of cases) to per-

A-STABLE BLOCK IMPLICIT ONE-STEP METHODS

259

form all computations exactly b y using rational arithmetic. The use of double precision floating point arithmetic on a CDC 6600 computer allows the computational scheme to proceed without error as long as the integers do not exceed 29 decimal digits. If one is careful about reducing fractions, this algorithm and this computer word length suffice to establish Lemma 7 conclusively. We now collect the major results of this section in T~EORE~ 3. Consider the r-block implicit one-step methods defined by the interpolatory formulas of Newton-Cotes type. A sufficient condition for the numerical scheme to be A-stable is that P ( - 2 ) be a stable polynomial. I n particular, for r = 1, 2 , . . . , 8 the corresponding block method is A-stable and convergent of order r + 1 for r odd and of order r + 2 for r even. For r = 9, 10 the method is not A-stable. 5. High order A-stable block methods. We shall now show how A-stable block methods of arbitrarily high orders of accuracy can be constructed. The essential idea will be that an acceptable r-block method is completely specified by any given polynomial P ( x ) = l +~r=laixi. Thus, taking a point of view opposite to the preceding section, we might hope that b y choosing certain polynomials P, the resulting block methods would yield rational functions Pr(,~)/P(~) satisfying the stability condition of Theorem 2. We shall construct A-stable methods using the diagonal and subdiagonal Pad6 approximations to erA since they have precisely the properties desired of Pr(~)/P(~). Using the analysis of section 3 and substituting (7) into (4), we see that after some manipulation the following identities in ~ arise: (15)

p(2)dk +jffii ~ pj(Z)bkj = Pk()~)2-P()O ,

~:= 1 , 2 , . . . , r .

For 2 = 0 the right hand member is to be taken as the limiting value. Let us now suppose we start with real polynomials Po*(2)=P*(2), P1"(2),. • • ,Pr*(2) of degree at most r which are linearly independent and normalized such that each Pj*(O)= 1. We shall further require that P*(2) be of exact degree r. Next, suppose we choose r + 1 distinct real values ~1,~2. . . . ,2~+1 and obtain B and d b y solving r systems of r + 1 linear equations of the t y p e given b y (15), which we refer to as (15") indicating the use o~ Pi*(2). That is, we solve the matrix systems W~k = vk for k = 1. . . . . r where W is the matrix whose j , l entry is P~_1(~i) and ~k is the vector whose j t h component is [Pk*(2j)-P*(2~)]/~j. B and are now determined b y setting (dk, bkl . . . . , bkr)T equal to 8k, the solution

260

l:t. A. W A T T S A N D L. F. S H A M P I N E

vector of the /cth system. This process is well defined since W is nonsingular b y virtue of the linear independence of the polynomials Pk*. In fact, because of the normalization property the right-hand member of (15") represents a polynomial of degree less than r so has a unique representation in terms of the basis set Po*(2) . . . . . P~*(2) for the r + 1 dimensional space of polynomials having degree at most r. Thus the above process uniquely determines B and d as candidates for an r-block method. For this procedure to be useful in what follows we need to guarantee that each Pt(2)=Pi*(2). B u t if we define (16)

Pk*(~) Yn P*(~)

Yn+k = - - - -

for ]c= 1 , . . . , r ,

then, except for a finite number of points at which P*(2)= 0, equations (15") m a y be used to obtain a system of the form (4). On the other hand, a unique solution of this system is given b y (7) for all values of 2 for which P(2) 40. Thus the rational functions of (7) and (16) represent the same functions at all but a finite number of points ~ hence are identical. While this is actually sufficient for our stability arguments, we can also conclude that the corresponding polynomials must be identical. The simple argument is detailed in [17]. In summary, the above constructive technique for obtaining an r-block method defined b y B and d does indeed produce the polynomials (5), (6) which are correspondingly identical to those of the starting set P*(2),...,P~*(2). Let us now see how we can obtain a satisfactory polynomial set. From the results of sections 3,4 (also see equations (12)), if all formulas of an r-block method have local accuracy at least O(h~+l), then (17)

Pk(a) - - - - = e k X + O ( ] 2 [ ~+1)

for k = l , 2 , . . . , r .

Thus, b y choosing a polynomial P(2) of exact degree r and normalized b y P ( 0 ) = 1, equations (17) then give us an explicit w a y of determining the polynomials P~(2); viz. the coefficients are given b y Lemma 3. We next need to establish the linear independence of such a set. LEMMA 8. Po(x)--P(x),Pl(x) . . . . ,Pr(x) are linearly independent on every interval [~,fi]. P~OOF. Suppose ~.~=oCkPk(x)----0. Substitution of the expressions for the Pk produces ~_~=0Vtxt- 0 where

,o=~ck, k=O

,i=~Ckakt k=O

fori=l,2 .... ,r.

A-STABLE

BLOCK IMPLICIT OI~E-STEB METHODS

261

B u t linear independence of 1,x . . . . . x ~ implies that all y i = 0 . Thus we obtain sequentially ~ % = 0 , Zk%=O, Zk~ck=O. B u t this is just Vr~=O where V is a Vandermonde matrix. Hence ~ = 0 and we have established linear independence. For convenience let us refer to the above block construction procedure as Method I. We shall prefer a mathematically equivalent, though computationalty different, procedure for obtaining B and d. B y asking about the accuracy of the formulas derived from Method I, we are led quite naturally to Method II, which we define b y the following theorem. T~no}cEM 4. Let us suppose that P(2)=~i=oai2 t, a 0 = l , ar=~0, is a given polynomial. For each j = 1, 2 . . . . ,r, determine B and d from

dj+ ~ bjk = j

(18)

/c=l

(19)

bjkk~-I - - j P k=l

for p -- 2 . . . . . r

P

(20)

bjkk r = - r ! k=l

at_k+ 1~..

k=l

Then B and 3 are uniquely determined and the resulting bloclc method has global accuracy at least O(h~). Furthermore, det ( I - 2 B ) is indeed the given polynomial P(~) and the P~(~) defined by (6) satisfy (17). PROOF. Note that the coefficient matrix for Method II is just the transpose of a Vandermonde matrix so it is clear that B and d are obtained uniquely. Noting that the degree of precision of the formulas, cf. section 4.1, requires each formula to have local truncation error O(hr+l), Theorem 1 then guarantees that the resulting block method is at least order r. Once we establish that d e t ( I - 2 B ) = P ( 2 ) we are done. We shall accomplish this b y showing that B and 3 computed from Method II are the same B and 3 computed from Method I. The previous analysis regarding Method I then allows us to conclude that the two polynomials in question are identical. Let us suppose that B* and d* are obtained b y Method I. Thus the given polynomial P(~) is used to generate the Pk(2) b y (17) and the construction procedure yields B*, d* satisfying P ( ~ ) - - d e t ( I - 2 B * ) , Pk(2)---det [Bk*(2)], and (21)

Pj(2) = 2 ~ b~Pk(Z)+P(2)[1 +~d~*] k=l

for j = 1,2 . . . . . r .

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H. A. WATTS

We now proceed to show form of (15),

B=B*

AND

and

L. F. SHAMPINE

d=d*.

First, rewriting (21) in the

r

d~*P(2) +]~ b~Pk(2 ) = j + O(121), k=l

and letting 2 -~ 0 in this equation, we see that B* and d* satisfy (18). Using this fact in (21) we get r

r

i--1

]fii-I

\

"

~k~=lb~{i~=l(Z~=o(i_l~.al)~i }

""

~ [~

ji-z a~_jar~r+1

Equating corresponding terms in 2 we evidently obtain

j i--l+l

r

i--1

( i - - / + 1)! a z

ki4

2 b;Z----al = k=l l=o(i--1)! --ja r

for

i=1,2 ....

for

i=r

,r-1

In a sequential manner B* and d* can now easily be seen to satisfy equations (19) and (20). B u t we have already asserted the uniqueness of the solution to (18), (19), (20). Hence B* = B , d* = d and we are finished. Thus, we have an easy, well-defined procedure for obtaining new formulas for acceptable r-block implicit methods simply b y choosing a desirable polynomial P(2) = 1 + ]~a¢2i and solving for B and d from equations (18), (19), (20). The required polynomial relations are then automatically satisfied. While no assumptions are being made about the zeros of P(2), our primary interest lies in A-stable methods and, accordingly, we shall insist that the zeros have positive real parts.

5.1. A-stable methods of arbitrary order. Let us now examine particular choices of P(2) which produce A-stable methods. The Pad6 approximation to ez is a rational function ~pq(g)/dpq(g) where n~q(Z) and dpq(z) are polynomials of degrees q and p, respectively, satisfying (22)

%q(Z) = dpq(Z)ez+O(lzl~+q+l)

as lzl-~ 0 .

Their explicit form is

q (23)

%~(z) d~q(z)

__

k=O

P

(p+q-k)v q~ .

.

zk

(p+q)! ~!(q_~)T (p+q--k)T p!

• ( - I) k Zk k=0 (p + q) ! k t(p -/~) !

A.STABLE

BLOCK

IMPLICIT

ONE-STEP

263

METHODS

Looking at the diagonal Padd entries, i.e., p = q = r , we see that n~(z)= dry(- z). Furthermore, it is known [3] that the zeros of d~(z) all lie in the right half plane Rez > 0 and tnrr(z)/d~r(z)l < 1 for all R e z < O. Hence we obtain the following lemma. LEMMA 9. Let an arbitrary r be given and set (24)

P(~.) = d,,(r~.) = 1+

/~=t

( - 1)~

2r(2r-1)

"'"

(2r-k+l)

"

Then the r-block implicit one-step method as constructed in Theorem 4 is A-stable. PROOF. We know that the resulting polynomial Pr(),), of degree at most r, satisfies (17) and, hence, is explicitly determined with respect to the given P(2) of (24). But also the rth degree polynomial n~r(rX) is uniquely determined from a similar relation (22). Necessarily, P~(2)= nr~(r~). The preceding comments and Theorem 2 now imply the property of A-stability. Next we shall examine the sub-diagonal Padd entries, i.e., p = r , q = r - 1 . Axelsson [2] has shown that all the zeros of drr_l(Z ) lie in the right-half plane Rez > 0 and that ]nrr_i(z)/drr_i(z)I < 1 for all Rez < 0, z 40. Thus in the same spirit as Lemma 9 we obtain LEMMA 10. Let an arbitrary r be given and set (25) P(~) = d,r_i(r2 ) = 1 + ~ ( - 1)k

(2r-- 1)(2r-- 2)

k=l

(2r-- k) "'"

Then the r-block method constructed from P(2) is A-stable. We have already seen that these new classes of formulas are at least O(h ~) accurate. Let us now show that both classes are in fact O(M+i) accurate (excluding the sub-diagonal Pad5 method for r = 1). THEOI%EM 5. Let r be given and consider the classes of block implicit one-step methods determined by the polynomials P(~) given in (24) or (25), the latter for r> 1 only. The corresponding r-block methods are A-stable and convergent with global error O(hr+i). PROOF. We shall show that the formula constructed for the endpoint, i.e., for Yn+r, is always the ordinary Newton-Cotes formula. Hence, we can take advantage of the convergence results of Theorem 1. For the last formula to be the corresponding Newton-Cotes formula, equation (11)

264

H . A. W A T T S A N D L. F. S t t A M P I N E

with j = r a n d p = r + 1 m u s t also be satisfied. B u t this is the same as requiring (13) to be satisfied with k=r. For the diagonal Pad6 block m e t h o d s determined b y the polynomials P(~) in (24) we have r ! ( r + k - 1)! r r-k+l at_k+ 1 = (-- 1)r-k+l (2r)! (/c--1)! (r--/c+ 1)! so t h a t the i d e n t i t y we wish to establish is, from (13), _-

/c

k=l

1

L (Tr)i

]

(r-t-l)!"

B u t a little manipulation gives the equivalent form (_l)r_k_

~=1

1

r+

/c

1

-1

= 0

"

W i t h an obvious change of variables this sum is evaluated as ( r +r 1 ) = 0 in the f o u r t h equation of p. 1t of [14]. A similar teelmique works for the sub-diagonal Pad6 block methods determined b y P(;t) in (25). We have, therefore, proven t h a t the last formula in the block scheme constructed from the denominator polynomial of the diagonal or sub-diagonal ( r > 1) Pad6 approximations to e r~ is indeed the Newton-Cotes formula. Application of Theorem 1 a n d L e m m a s 9,10 now shows the m e t h o d s are A-stable a n d of order r + 1. Explicit formulas for the Newton-Cotes block m e t h o d s (for r = 1,2, . . . . 8) a n d the diagonal and sub-diagonal Pad6 block methods (for r = 1 , 2 , . . . , 5 ) are presented in [17]. 5.2. The concept of stiff stability. I n studying A-stability we have seen (section 3), for n = m r a n d

re=O, 1,..., Yn+i = tjr(~h) [ - ~ h ~ - ]

,

J = 1. . . . . r

where we have applied the block m e t h o d to the differential equation (3). Absolute stability requires t h a t [Y~+¢Ido n o t increase as n -~ ~ for all h > 0 with R e ~ < 0. While the interpolatory Newton-Cotes formulas (for r = 1,2 . . . . . 8) a n d the diagonal Padd block methods are b o t h A-stable, [Pr(~h)/P(o~h)] < 1, we note t h a t [Pr(ah)/P(ah)] -~ 1 as h R e ~ -~ - ~ . Thus, when - h Re~>> 1, we find t h a t the numerical solution does n o t exhibit the rapidly decaying behavior of the true solution. Furthermore,

A-STABLE BLOCK IMPLICIT ONE-STEP METHODS

265

for odd values of r we see that P r ( a h ) / P ( ~ h ) - ~ - 1 . For - h R e a > ~ 1 we now have a slowly damped b u t rapidly oscillating solution. Such behavior is frequently undesirable. These remarks seem quite pertinent when discussing the solution of "stiff" systems of ordinary differential equations (see Gear [12]). A constant coefficient system u' = S u is said to be stiff if all the eigenvalues of the matrix S are in the left half-plane Rez < 0 and, moreover, their real parts differ greatly in magnitude. That is, if #1,- •., #~ are the eigenvalues of S, we are supposing that Re/~l__> 1. The numerical solution of such systems usually requires very small steps to stay within the stability region of the particular method being used. Of course, A-stable methods do not have this limitation and herein lies their value. If in addition - h Re#l>>l, the true solution will have a component like e"I~ which rapidly looses its significance and, accordingly, it is desirable for our computed solution to behave in the same manner. From our previous remarks, the diagonal Padd block methods and the Newton-Cotes block methods do not exhibit this desirable behavior. On the other hand, we note that the sub-diagonal Padd block methods have the property that IP~(ah)/P(ah)[ -+ 0 as h R e a -+ - ~ . Such methods are termed "stiffly" A-stable, see Axelsson [2]. Perhaps "strongly" A-stable is a better terminology since Gear refers to his specially designed techniques as stiffly stable. This additional property should prove valuable for numerical purposes in solving stiff equations.

REFERENCES I. O. Axelsson, Global integration of differential equations through Zobatto quadrature, BIT 4 (1964), 69-86. 2. O. Axelsson, A class of A.stable methods, BIT 9 (1969), 185-199. 3. G. Birkhoff and R. S. Varga, Discretization errors for well-set Cauchy problems I, g. of Math. and Physics 45 (1965), 1-23. 4. J . C. Butcher, Implicit Runge-Kutt~ processes, Math. Comp. 18 (1964), 50-64. 5. J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp. 18 (1964), 233-244. 6. J. C6a, Equations differentielles methode d'approxixaation discrete p-lmplicite, Chiffres 8, No. 3 (1965), 179-194. 7. F. Ceschino and J. K u n t z m a n n , Numerical Solution of Initial Value Problems, Prentice--Hall, Englewood Cliffs, New York, 1966. 8. L. Collatz, The Numerical Treatment of Differential Equations, Springer-Verlag, New York, 1966. 9. W. Coppet, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965.

266

H. A. WATTS AND L. F. SHAMPINE

1O. G. Dahlquist, A special stability problem for linear multietep methods, BIT 3 (1963), 27-43. 11. B. L. Ehle, High order A-stable methods for the numerical solution of systems of D.E.s, BIT 8 (1968), 276-278. 12. C. W. Gear, The automatic integration of stiff ordinary differential equations, I F I P Congress, Edinburgh, 1968, 187-193. 13. W. E. Milne, Numerical Solution of Dif]erential Equations, Wiley, New York, 1953. 14. J. Riordan, Combinatorial Identities, Wiley, New York, 1968. 15. J. B. Rosser, A Runge-Kuttafor all seasons, SIAM Rev. 9 (1967), 417-452. 16. L. F. Shampine and H. A, Watts, Block implicit one-step methods, Math. Comp., 23 (1969), 731-740. 17. H. A. Watts, A-stable block implicit one-step methods, P h . D . dissertation, University of New Mexico 1971, also available as Sandia Laboratories report SC-RR-71 0296. 18. K. WTright, Some relationships between, implicit Runge-Kutta, Collocation and Lanczos ~: methods, and their stability properties, BIT 10 (1970), 217-227.

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