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UIUCDCS-R-73-570 
 
 C00-1U69-0220 
 
 ?/ 
 
 THE EFFECT OF VARIABLE MESH SIZE ON 
 THE STABILITY OF MULTISTEP METHODS 
 
 C. W. Gear 
 K. W. Tu 
 
 May 
 
 U 
 
 1973 
 
 April 1973 
 
UIUCDCS-R-73-570 
 
 THE EFFECT OF VARIABLE MESH SIZE ON 
 THE STABILITY OF MULTISTEP METHODS* 
 
 ty 
 
 C. W. Gear 
 K. W. Tu 
 
 April 1973 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 
 UNIVERSITY OF ILLINOIS AT URBAN A- CHAMPAIGN 
 
 URBAN A, ILLINOIS 6l801 
 
 * Supported in part by grant US AEC AT( 11-1 ) 1^69. 
 
1 . INTRODUCTION 
 
 This paper is concerned with the stability over the finite 
 interval ts[0,T] of multistep formulas subject to varying step sizes. 
 Although practical algorithms vary the step size in order to use as 
 large a step as possible consistent with a local control of the estimated 
 error, the existing literature — with the exceptions noted below — deals 
 only with the stability of fixed step formulas. 
 
 The standard derivations of multistep formula, e.g. Henrici [3], 
 are based on equal intervals. There are two common ways of handling 
 variable steps for multistep formula. One is to interpolate through the 
 computed points to get values at equally spaced points for use in a 
 standard equal interval formula. The other is to derive multistep formula 
 directly based on unequal intervals . These techniques do not have the same 
 stability characteristics. In 1962 Nordsieck [k] noted an apparent 
 instability in his interpolation version of Adams formula if the step size 
 was varied too frequently. In 1969 Piowtrowski [5] proved that a variable 
 step form of the Adams-Moulton formula was stable and convergent. Brayton, 
 et.al. [l] and Gear [2] (pp. 1U5-1U6) point out that the two techniques 
 for changing step size are not equivalent, and Brayton, et.al. give test 
 examples that show that both can cause instability, but that the technique 
 of step variation used by Piotrowski might be more stable. 
 
 The stability of a method is affected by three factors: the 
 underlying multistep formula, the technique used to handle variable 
 steps, and the scheme used to select step sizes — the formula, technique, 
 and scheme for short. Method will be used to refer to the combination 
 of a formula and a technique. This paper introduces the idea of a step 
 selection scheme, and defines the stability and convergence of a method 
 
with respect to a scheme. A stability condition is derived. It is 
 necessary for stability, and, together with consistency, implies 
 convergence and stability. 
 
 It is easy to generate consistent formulas. The more difficult 
 task is to determine whether a combination of formula, technique, and 
 scheme is stable. The two common techniques for step changing are 
 examined, and it is proved that the variable step technique is superior 
 to the interpolation technique for Adams formula. Specifically, it is 
 proved that the variable step technique used with Adams formula is 
 stable and convergent with respect to any step selection scheme, while 
 the interpolation method can be unstable if the steps are allowed to 
 change too much. However, it is proved that the interpolation technique 
 with the k-step Adams formula is stable if the step selection scheme 
 guarantees that the step is constant for at least k-steps after each change 
 
 The investigation is extended to all multistep formulas, whether 
 explicit, implicit, or predictor-corrector. It is proved that if the 
 underlying formula is stable, then both step changing techniques lead to 
 stable methods if the rate of change of the step size is small. It is 
 also shown that common error control algorithms lead to small rates of 
 change in the step size. 
 
2 . BACKGROUND 
 
 The two step changing techniques will be called interpolation 
 and variable step, respectively. Intuitively, we define them as follows. 
 Suppose we are using a k-step formula for integration and suppose that 
 an approximation y . to the solution is known at k values of the 
 independent variable t, say t ., k > i _> 0. The interpolation 
 technique for changing step size to h consists of the following two steps: 
 
 1. Interpolate through the known points to get the values 
 
 of the solution at the points t .=t -ih,k>i>0. 
 
 n-i n — 
 
 2. Use the fixed step integration formula on these new 
 
 points t . to find the value at t -, = t + h. 
 n-i n+1 n 
 
 In the variable step technique we start with k unequally spaced 
 points t . , k > i >_ and compute the coefficients of the multistep formula 
 
 Vl = "l, n ^n + - • - + a k,n y n-k + l + B 0,n \ Cn (l) 
 
 + B l,n h „-l y n + -'- +6 k,n h n-^n-k + l 
 
 so that it is exact to the required order, where h = t _ - t . Since 
 
 n n+1 n 
 
 the order cannot exceed 2[k/2]+2 if the formula is to be stable for fixed 
 step sizes, additional restrictions must be imposed. These consist of 
 specifying the values of some of the coefficients a and g. The fixed step 
 formula underlying the variable step technique is said to be the formula 
 that is obtained from (l) when equally spaced points t . are used. For 
 example, if we required that a. = 0, j >_ 2, and that (l) have order k+1, 
 then Adams-Moulton is the underlying fixed step formula. If we also ask 
 that g =0 and lower the order to k, we get a variable step method based 
 on the Adams -Bashforth formula. 
 
The stability of an integration formula subject to a step 
 changing technique may depend on the way in which the steps are varied. 
 Therefore, we must define stability with respect to a particular step 
 selection scheme. First, therefore, we state 
 
 Definition 1 A step selection scheme is a function 9 such that 
 
 h = hQ(t } h) 
 n n 
 
 where, for all h > 3 < t < T 
 
 1 > Q(tjh) 1 A > 
 
 NOTE: As h is reduced to zero, the maximum step size is reduced to zero. 
 The lower bound A serves to prevent zero step sizes for any non- 
 zero h; these would not give useful numerical integration schemes! 
 In this paper we will be concerned only with differential 
 equations y' = f(y,t) for which f(y,t) is Lipschitz continuous in the 
 strip |y| <°°, <_ t <_ T < °°. Small perturbations to the solutions of 
 such equations at a given point cause small perturbations later on. 
 
 Stability means that small numerical disturbances at one point t cause 
 
 n 
 
 small perturbations in the numerical solution at later times for all 
 small h. Convergence means that the numerical solution can be made 
 arbitrarily accurate ash+0. 
 
 Definition 2 A method is stable with respect to a scheme 6 if there 
 
 exists a constant M < °° (dependent on the differential equation only) such that 
 
 \y -ii < M \y - ij 
 '^m v m> li7 n a n ] 
 
 for all <_ t < t <T 3 where y . and y. are two numerical solutions. 
 
 I I III Is (s 
 
Definition 3 A method is convergent with respect to a scheme 8 if the 
 computed solution y converges to 
 the starting errors tend to zero. 
 
 computed solution y converges to y(t ) for any <_ t <_T as h ■*■ and 
 
 We 'will only discuss a single differential equation. The 
 extension to systems of equations is straightforward, although notationally 
 complex. We will also assume that f(y,t) and y(t) are differentiate 
 as. often as necessary to save stating all the conditions each time. The 
 results are valid if f is only Lipschitz continuous , and can he obtained 
 "by replacing f = 3f/9y by the Lipschitz constant L whenever f appears. 
 
3. NOTATIONAL DEVELOPMENT 
 
 By developing a uniform notation for all the formulas and step 
 
 changing techniques to be discussed, we will be able to give general 
 
 proofs in Section h that can be applied to all cases . Some of this 
 
 notation has been developed in [2], It is summarized briefly here and 
 
 suitably extended. 
 
 Consider the integration formula (l) with 3„ =0. It is an 
 
 0,n 
 
 explicit formula which can be used directly. It can also be used as a 
 predictor for a corrector formula. We will write the corrector formula as 
 
 *M ] = ^n ^n + -" + «k,n ^n-k + l + g S,n h n f( Ci> W 
 
 + (B* h . y' +...+ g* h , y« ... 
 
 l,n n-1 n k,n n-k n-k+1 
 
 (2) 
 
 .(0) 
 
 A predictor-corrector formula is one in which a predicted value y is 
 
 computed by the explicit equation 
 
 .(0) 
 
 y'n=ot, y + . . .+ a y , , 
 J n+1 l,n °n k ,n ^n-k+l 
 
 (3) 
 
 + 3 n h y' +...+ e h y' ^ 
 l,n n-1 n k,n n-k n-k+1 
 
 and the equation (2) is used to compute a fixed finite number of corrected 
 
 (l) (M) (m) 
 
 values y ,.,..., y '. The approximation y ,. is defined as y , ' and y',. 
 n+1 n+1 n+1 n+1 n+1 
 
 is set to either f (y ) or f(y ) depending on whether or not a final 
 function evaluation is needed. 
 
 An implicit formula is one in which equation (2) is solved for 
 y = y . One way of doing this is to iterate a predictor-corrector 
 formula until successive iterates are approximately equal. This converges 
 for h sufficiently small but may diverge for large h. Other methods, such 
 as Newton-Raphson, can also be used to solve (2). The stability of a 
 method based on an implicit formula is independent of the process used to 
 solve equation (2), so we do not need to specify that process. 
 
During the computation, a number of previous values of the 
 
 function y . and the scaled derivatives h . _ y 1 . must be saved, 
 'n-i n-i-1 n-i 
 
 Let us collect these together in a column vector y defined "by 
 
 ^n = [ V y n-l : 
 
 » K , y 1 , K o y ' -. » • • • » * t. y ' ^ ] 
 
 T 
 
 n-k+1' n-1 J n* n-2 J n-1 
 
 n-k ' n-k+1 - 
 
 for a k-step formula, where T is the transpose operator. (if some of 
 these values are not needed, we can drop them from the vector. For 
 
 example, in Adams formula, y would be [y , h _ y ',..., h . y' . ,. 
 
 *-n n n-1 n n-k n-k+1 
 
 A numerical integration method is a method for computing 
 
 ] T .) 
 
 y from y . We will first describe it for the variable step technique, 
 
 Let y\,i be defined as 
 
 r (m) 
 
 y:i? } > K n y:.-.-» 
 
 n+1' *&""* °n-k+2' n J n+1 ' n-1 J n 
 
 where h y'*? is h f(y ( ™, , t .. ) for m > 1, 
 
 n n+1 n w n+l n+1 — 
 
 y' 1 
 
 n-k+1 <y n-k+2 J 
 
 h n y n+l Y l,n y n + Y 2,n y n-l + -'- + \ >n y n -k+l 
 
 + 6 h v' + 5 h y' +...+ 6 h y' 
 
 l,n n-1 ^n 2,n n-2 °n-l k,n n-k ^ n-k+1 
 
 when m = 0, and the coefficients y and 6 are given by 
 
 Y- = (a. - a* )/6* 
 i ,n i ,n i ,n 0,n 
 
 and 
 
 6. = (6. - (3* )/6* 
 
 i,n l ,n i,n ,n 
 
 From these definitions we can write 
 
 til - * n *, 
 
 (h) 
 
where A = 
 n 
 
 — 
 
 
 
 
 
 
 
 
 — 
 
 l,n 
 
 2,n 
 
 • • • 
 
 k-l,n 
 
 k,n j l,n 
 
 B 2,n 
 
 • • • 
 
 e k-l,n 
 
 6 k,n 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 •• 
 
 • 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 l,n 
 
 Y 2,n 
 
 9 ■ • 
 
 Y k-l,n 
 
 Y ! 5 
 
 k,n j l,n 
 
 6 2,n 
 
 
 6 k-l s n 
 
 k,n 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 B • • 
 
 
 
 
 
 
 
 
 
 • 
 
 • • 
 
 • • • 
 
 
 • 
 
 . 
 
 
 
 
 i o 
 
 
 
 
 1 
 
 
 
 J 
 
 and 
 
 (m+l) (m) r f( y (m) , t ) - h y' U) ] 
 ^n+1 ^n+1 V n K7 n+V n+r n y n+l J 
 
 (5) 
 
 where 
 
 * = t3* , 0,..., 0, 1, 0,..., 
 
 (6) 
 
 Equation {h) represents the predictor — the evaluation of 
 
 y ,, and h y' Ln as linear combinations of the known information plus 
 "n+1 n J n+1 ^ 
 
 the shifting over of the saved information and the discarding of y 
 
 and y' . Equation (5) represents the corrector — the evaluation of 
 
 s-t ( m ) a. \ a j.-, n ^ • j> (m.) ^ •, ,(m) _ 
 
 £\Y xD "t , -, ) and the updating of y ' and h y ',. . For convenience, 
 n+1 n+1 n+1 n n+1 
 
 we define 
 
 / (m) x . _/ (m) , >. . , (m) 
 
 V^n+1 
 
 n+1' n+1' 
 
 n "n+1 
 
 (T) 
 
Then equation (5) can "be written 
 
 (m+l) (a) +l F ( (m) } (8) 
 
 ^+1 *n+l -n n v,L n+l ; v ; 
 
 If a predictor only formula is used, we still formally must apply one 
 
 step of (8) with H = £_ = [0,. . ., Q, 1, 0,,..,0] . (in this case the 
 
 value of a* and 3* are unimportant.) If a final function evaluation is 
 
 used, we apply (8) M times using the & given in (6), and then apply 
 
 p 
 one further step of (8) using & = l_ . 
 
 Now let us consider a fixed step method in preparation for 
 
 formalizing the interpolation technique. In this case, the a, 3, y, and 
 
 6 in A and I are independent of n. Suppose we have just completed the 
 
 step from t . to t using step h . . The values saved in y are the 
 n-1 n n-1 •4i 
 
 computed approximations at the points t -ih , <_ i < k. Before 
 
 stepping to t , , = t + h we must interpolate to the points 
 ** n+1 n n r 
 
 t . = t - ih . This is a linear process and can he represented by 
 n-i n n 
 
 first premultiplying y by an interpolating matrix C . Consequently, 
 we write the predictor step for the interpolation process as 
 
 C\ - A C n *n (9) 
 
 The corrector step (8) is unchanged except that I = l_ is 
 independent of n. 
 
 Both for theoretical development and computational ease, it is 
 convenient to perform a transformation based on Nordsieck's [k] form of 
 Adams formula for the interpolation technique. This consists of computing 
 and saving Q y rather than y where Q is a nonsingular matrix. The 
 particular form of Q used is obtained as follows: The set of values of 
 y _. and y'_. <_ i < k determine a unique (2k-l)-th degree polynomial. 
 
10 
 
 Represent that polynomial instead by its value at t and the values of 
 its first 2k-l scaled derivatives h J y /j ! , 1 <_ j < 2k. (This 
 leads to what is called a 2k -value method in Gear [2].) If y is the 
 
 set of saved values of y and hy' "based on step h 
 
 n-1 1 
 
 and a is the set 
 — n 
 
 (10) 
 (11) 
 
 of values of [y , h . y', h 2 _ y"/2,..., h 2 *" 1 y (2k-l) /(2k-l) ! ] T 
 n n-1 n n-1 n n-1 n 
 
 representing the same polynomial of degree 2k-l, then Q is defined 
 "by a = Q y • Q is independent of y and h. . If we apply this 
 transformation to (9) and (8) we get 
 
 Q 4°| = (QAQ" 1 )(QC n Q~ 1 )Q y^ 
 
 To simplify the presentation we will write y rather than a , A rather 
 
 than GAQ , C rather than QC Q , I rather than Q£ , and F(c) rather 
 n n — n -n — 
 
 than F(Q c_) when we are talking about the transformed scheme. 
 Consequently, equations (10) and (ll) are identical to (9) and (8). 
 The advantage of making the transformation is that 
 
 pv_1 
 C n = diag[l, h n /h n _ l5 ..., (\/\_ ± ) 1 (12) 
 
 so that the interpolation process is simplified. This was Nordsieck's 
 
 motivation. Finally, we note that (9) is identical to (k) if A is 
 
 n 
 
 interpreted as A C for the interpolation technique. Consequently, 
 equations (k) and (8) represent the predictor only formula or the 
 prediction-correction formula in either step changing technique. 
 
 The corrector only formula can sometimes be implemented by 
 iterating the corrector to convergence and treated theoretically by 
 setting y = lim yV , • However, this process may not converge (as 
 in stiff equations), in which case the corrector must be solved by other 
 
11 
 
 means. We note that the corrector equation (8) always adds on a 
 multiple of & to the current value. Consequently, the final value 
 has the form 
 
 Vl ■ J&i + "4 = A n *n * «*» (13) 
 
 where W is a scalar. The value of go is such that 
 
 nzlH * co^) = o (no 
 
 (Although we have used predictor coefficients in A , some straightforward 
 arithmetic will show that the final value of y is independent of these 
 coefficients when the corrector equation is solved by (13) and (lU).) 
 
 We must now develop the error propagation formula for equations 
 (k) and either (8) or (13) and (lU). The global error n is defined by 
 
 * n " ^ - £(t a ) (15) 
 
 where y_(t ) is the value of the components of the vector y as 
 evaluated along the desired solution of the differential equation. 
 
 The truncation error d is the error introduced by one step of the 
 
 -n 
 
 method, starting from the solution of the differential equation. Thus, 
 d is defined by the equations 
 
 either 
 
 &? - tit ♦ 4 F nO - 
 
 ^n+1 ^ti+I 
 
 if M corrector iterations are used, or 
 
 2^4.t = liVi + wi = A y_(t ) + Sd , (16) 
 
 *-n+l *-n+l — n n *- n —a 
 
12 
 
 where F(jf _ ) = if the corrector is solved exactly, and 
 
 4 = 4 +1 - 2L(t n+1 ) 
 
 From the definition of e , , and d 
 
 -n+1 -11 
 
 ^n+1 = 4+1 " 1.(^+1 ) 
 
 " 4+1 " 4+1 + 4+1 " ^ n+r 
 
 = 2,,+! " 4+1 + 4 (1T) 
 
 (Note that if round-off errors are to "be considered, they can be included 
 in d at this point, so that the computation of y _ can be considered 
 to be exact.) If M corrector iterations are used, (17) can be written as 
 
 -n+1 ^+1 *ia+l -n 
 
 =e i) + ^ve i) )-e i) -4vai i, >^ 
 
 3F (M) 
 by the mean value theorem. - — is the row vector of partial derivatives 
 
 of F with respect to the components of y_ evaluated somewhere in the 
 
 interval (y , f ). From equation (7) we see that 
 
 |^ M - [h f< M) , o,..., 0, -1, 0,..., 0] 
 dy_ n y 
 
 where r M ' is 9f/3y evaluated in the interval (y^~ , y +~ ) . The 
 
 process leading to (l8) can be repeated to get 
 
 ,(M) _(D 
 
 ^n+1 
 
 -n 3y_ -n 3y_ "-n+l *-n+l -n 
 
 aF (M) . (1) 
 
 - [I +£ I 1 ]...[! +£ I 1 1 A e + d (19) 
 
 -n 3y_ — n 3y_ n -n n 
 
We write this in the form 
 
 13 
 
 e Ln = S e + d (20) 
 
 -n+1 n -n -n 
 
 T 
 Let e. be the unit row vector consisting of a one in the i-th 
 —i 
 
 T 
 position and zeros elsewhere. Thus, e_ is [l, 0,..., 0]. Let 
 
 subscript d represent the component corresponding to the h j l 
 
 entry in y . (d is k+1 for the untransformed method, 2 for the 
 
 transformed method.) Then 
 
 fl (m) . h f <»> J . £ ' ( 21) 
 
 9v_ n y — 1 -d 
 
 Therefore, from (19), (20), and (21) 
 
 's = n [(I -I e?) + h f (m) I e?]A (22) 
 
 n . -n-d ny-n— In 
 
 m=l 
 
 T M 
 = [(I - I e, ) + I {Products of M terms of form 
 —n-d 
 
 h f (m) I e%r (I - I eT) 
 n y -n — 1 -n — d 
 
 with at least one of the former} ]A 
 
 n 
 
 We define 
 
 Then we can write (22) as 
 
 / T.M . 
 
 S = (I - I e, u A 23 
 
 n -n — d n 
 
 S = S + h S (21+) 
 
 n n n n 
 
 where S is a matrix whose elements are polynomials in h , f and 
 n n y 
 
 entries from A and % . 
 n — n 
 
Ik 
 
 Lemma 1 If a, g, a* and 3* are uniformly "bounded or if A and I 
 n -n 
 
 are uniformly "bounded so is j |S | j for all |h| <_ u. 
 
 Proof The elements of S are polynomials m h whose coefficients 
 _ n n 
 
 are polynomials in f and the entries of A and I . If I is bounded 
 
 y n -n y ' 
 
 by the Lipschitz constant, while entries in A and I , except possibly 
 
 n — n 
 
 for the d-th row of A , are bounded if a, 3, a* and 3* are bounded. The 
 
 n 
 
 d-th row of A contains elements of the form (a-a*)/3* , and could be 
 n ,n 
 
 unbounded if 3* can approach 0. However, from (22) and (2U), the only 
 
 way the d-th row of A enters S is when A is premultiplied by 
 * n n n 
 
 T 
 (I - i e ., ) which zeros the d-th row and changes the first row from 
 — n -d 
 
 a to a*. Consequently, the coefficients of h in S are bounded, so for 
 
 * n n 
 
 any u > 0, Is ; is bounded for all 0<h<u. If A and I are 
 j •> i i n i i _ _ n -n 
 
 uniformly bounded, the result follows similarly. 
 Q.E.D. 
 
 The behavior of the error e is determined by equation (20). 
 
 -n 
 
 When I Is is bounded, we can use equation (2U) to relate the stability 
 n 
 
 of the method to the properties of S . The following condition will be 
 
 shown to be necessary for a method to the stable: 
 
15 
 
 Definition 4 A method satisfies the stability condition with respect 
 to a step selection scheme if there exists a C < <*> 3 such that 
 
 n m-1 m-2 n 
 
 satisfies 
 
 | 1^1 | <_C (26) 
 
 n' 
 
 for all < t < t < T. 
 J — n m — 
 
 If an implicit method is used, the corrector is solved 
 
 exactly, and we must derive equations (20) and {2k) by a different 
 
 technique. Substitute (13) and (l6) in (17) to get 
 
 e n = A e + (w-fiS) i + d (27 
 
 -n+1 n — n — -n 
 
 (The subscript n has been omitted from l_ for clarity later.) We know 
 
 = F(v ii+1 ) - F(2n+1 ) 
 
 9v_ ^+1 *n+l' 
 3F 
 
 3v _ <A n ^ + («-a) D 
 
 Hence , 
 
 (u-fi) = -C|Jl> #A e } 
 
 3v_ — 3v_ n -n 
 
 Note that both quantities enclosed in braces are scalars. Substituting 
 in (27) we get 
 
 S„+i = A e " ^^> l <■¥■ A £ > + d 
 -n+1 nn 9 y_ — — 3v_ n -n -n 
 
 °%_ ~ - " ov_ n -n -n 
 
 = S e + d 
 n — n — n 
 
16 
 
 where 
 
 n 9y_ - - 3y_ n 
 
 i- [I - "'{L h f - a}" 1 £(h f e^ - e^)] A 
 1 n y d — n y — 1 -d /J n 
 
 Here, I . is the i-th component of l_. 
 
 If we choose u such that for 0<h < u, h JL f - £ is hounded away 
 
 n — n 1 y d J 
 
 from zero {l is one in the methods discussed), then we can write 
 |h n'l f y "V" laS ^f {1 +l l h n f y A d + °( h n>>- He ^, 
 
 s n = [I-^Ajg] A n + h n s n ' ( 28 ) 
 
 = S + h S 
 n n n 
 
 The form of S is the same as the previous form with M = 1 and I scaled 
 n — 
 
 so that Jl = 1, as it always is in methods discussed in this paper. 
 We also need to extend Lemma 1 to this case. 
 
 Lemma la || S || defined in (28) is hounded if a, 6, a* and B* 
 
 or A and I are uniformly bounded, 
 n -n 
 
 Proof From (28) and the preceding equation 
 
 S = -U n h f -I A' 1 f I e. A 
 n lnyd y In 
 
 + [U.h f -4 J" 1 + C" 1 ]^ eT A /h 
 lnyd M dnn 
 
 = -U n h f -2, J -1 f I e? A 
 lnyd y In 
 
 + nT^H-h f 4)" 1 f a s, e^ A 
 d lnyd y— 1-dn 
 
IT 
 
 All "but possibly the d-th row of A are bounded. The first term does 
 
 not involve the d-th row of A , and is, therefore, bounded for 
 
 n 
 
 < h < u. The second term involves the d-th row of A , but scales it by 
 — — n 
 
 £ ., = 3* . Hence, it is also bounded for < h < u. Hence, Is is 
 1 0,n — — ' ' n 1 ' 
 
 bounded. If A and I are bounded, the result follows directly, 
 n — n 
 
 Q.E.D. 
 
 
18 
 
 h. STABILITY AND CONVERGENCE 
 
 Theorem 1 If a method is stable with respect to a step selection 
 
 scheme 6j it satisfies the stability condition (26) with respect to 
 that scheme. 
 
 Proof Consider the differential equation y' = for which S , 
 n 
 
 defined "by (2k), is zero. Evidently, the difference between two 
 numerical solutions satisfies 
 
 ^n+1 ~ -n+1 ~ S n^n " Za) 
 
 i i A m i i 
 If the S II are not uniformly bounded, there exist t and t and 
 
 1 ' n 1 ' m n 
 
 bounded y - y such that 
 — n — n 
 
 y m ~ J m = S m (y - y ) 
 
 is arbitrarily large, implying that the method is not stable with 
 respect to 0. Hence stability implies the stability condition. 
 Q.E.D. 
 
 The convergence theorem depends on the following lemma. 
 
 Lemma 2 If, for the difference equation, 
 
 x^=Sx+hSx+hA (29) 
 
 —n+1 n -n n n -n n — n 
 
 there exist constants k , k , and k such that 
 
19 
 
 Is | | < k. ¥ n 
 n — 1 
 
 A < v ¥ n, 
 —n — 2 
 
 and | |s m | | < k. ¥ m > n 
 
 n — — 
 
 where 
 
 s n = I 
 
 n 
 
 and S is defined by equation (25) a 
 n 
 
 then, if k > 
 
 or if k =0 
 
 I II rv II II k2 i Vl ( W k 2 ,, M 
 
 1^11 1^ M^ll + -] e -- (30) 
 
 1^1 1 i^o H*qII +v 2 ( V t o ) (31) 
 
 Proof From (29) we have 
 
 N-l 
 
 x= s' x^ + E s 4.1 h S x + A 
 
 — N -H) „ n+1 n n — n -n 
 n=0 
 
 Therefore , 
 
 112,11 ik I I*, I I ♦ I Vn (k l 1 1^1 1 + k 2> (32 
 
 n=0 
 
 Consider the case k > first. We show (30) by induction. It is 
 evidently true for N = since k >_ |s | = 1. Assuming its validity 
 
 for n < N, we substitute in (32) to get 
 
20 
 
 N-l k„ k n k n (t -t A ) 
 
 I I xl I < k 11x11+ E kkh fk llxll+— I*- 01 n ° 
 I l%ll ±k Q | IXqI I + L K Q K n n LK Q HXqI I + k J e 
 
 n=0 1 
 
 k N-l k k (t -tj k„ 
 
 n II II 2 irn j. v i i v. 1 n t 2 
 
 = [k Q MxqII + — ][l + E Wn e ] - — 
 
 1 n=0 1 
 
 k N-l k n k,h k_k n (t -tj k 
 
 ^ n II ii . 2 T r _, _,/01n_x01 v n0 / 1 2 
 
 1 n=0 1 
 
 This completes the inductive proof of (30). If k = 0, equation (31) 
 
 follows from (32) directly. 
 
 Q.E.D. 
 
 The convergence theorem is now straightforward. 
 
 Theorem 2 If a method satisfies the stability condition with respect 
 
 to a step selection scheme 0, if a, a*. 3 and 6* or A and I are uniformly 
 
 bounded, and if the truncation error d satisfies 
 
 3 J — n 
 
 I Id I I < h e(h) V n 
 i \_ n \ i _ n 
 
 where e(h) + as h -*■ 3 then the method is stable and converges with 
 respect to <d 3 and there exist constants k~ and k. such that the global 
 error e_^ satisfies 
 
 llf^ll £k 3 ll^ll + k 4 e(h) for <_t N <_T 
 
 Notes: 1. The condition on d is a consistency condition. If the 
 -n 
 
 method has order r, then e(h) = 0(h ). However, a more useful way of 
 
 looking at this result is that if we control the step size h so that the 
 
 n 
 
21 
 
 truncation error is "bounded by eh , the global error is bounded by 
 
 k i |e | j + ki e, so that changes in the truncation error control are 
 
 proportionately reflected in the global error bound. 
 
 2. The condition that a, a*, 3 and $* or A and I be uniformly 
 
 n -n 
 
 bounded is also a form of consistency. It rules out the following type of case 
 The explicit formula 
 
 1 
 
 n+1 h „-h . 
 n-2 n-1 
 
 (h +h J 2 -h 2 . (h +h J 2 -h 2 . 
 
 n n-1 n-2 n n-1 n-1 
 
 h +h _ ~ y n h +h . " J n -2 
 
 n-2 n-1 n-2 n-1 
 
 h (h +h +h _) 
 n n n-1 n-2 
 
 h n-2 ****■ 
 
 is second order provided that h _ ^ h . . However, as h _ ■*■ h . , the 
 
 n-2 n-1 n-2 n-1 
 
 coefficients and the error term blow up. If we used this formula with 
 
 a step selection scheme which kept h _/h . bounded away from one, we 
 
 n-2 n-1 
 
 would get convergence if the method was stable, but if the step selection 
 
 scheme let h . /h „ •* 1 as h -*- , we may not get convergence. 
 n-1 n-2 
 
 Proof From (20) and (2U) the error equation for a method is given by 
 
 e ... = S g + d (33) 
 
 -n+1 n -n — n 
 
 ■ S e +h S e +d 
 n — n n n — n — n 
 
 Consistency and Lemma 1 imply that |s | <^ k. for some k < °° provided 
 
 that h <_ u. Lemma 2 can be applied to (33) with k = e(h) since the 
 
 stability hypothesis guarantees the existence of a constant 
 
 k^ such that I S II < k^ . 
 ' ' n ' ' — 
 
22 
 
 Therefore, 
 
 1 k 3 I 1% I I + \ e (h) v ° 1 t N 1 T 
 Hence, e^ J | -*■ if | e^ | and e(h) -*■ and we have convergence. 
 
 — Vi — 
 
 The difference between two numerical solutions satisfies 
 
 ^n+1 " 4+1 = S n (y n " V 
 The same analysis (with e(h) = 0) yields 
 
 Therefore, the method is stable. 
 Q.E.D. 
 
 The difficult problem left is to verify the stability condition 
 for various methods and step selection schemes. First, we will deal with 
 the Adams methods in Theorems 3, h, 5, and 6. Theorem 5 is an extension 
 of the result of Piotrowski [5]. 
 
 Theorem 3 A and i are bounded for the interpolation technique 
 
 using a step selection scheme 0. 
 
 Proof For the interpolation technique I = i_ is independent of n, and 
 
 is, therefore, bounded. A = A C where C , given by equation (12), is 
 
 n n n 
 
 — 2k+l 
 bounded by A " from definition 1. Therefore, A is bounded. 
 
 n 
 
 Q.E.D. 
 
23 
 
 Theorem 4 a, a*, 3 and 3* are bounded for the variable step Adams 
 
 method using a step selection scheme 9. 
 
 Proof The variable step Adams method has the form 
 
 y = v + R* h v' +...+ 3* h v 1 (35) 
 
 J n+1 *n P 0,n n ^n+l P k,n n-k ^n-k+l u?i 
 
 where the 3* are determined from the requirement that the method 
 i,n 
 
 have order k+1. (For an explicit method, 3* =0 and the order is k.) 
 
 ,n 
 
 A necessary and sufficient condition for the order to he k+1 is that 
 
 (35) he exact for the functions y(t) = (t - t ) , <_ r <_ k+1. This 
 
 leads to a system of k+1 linear equations for the 3* , namely 
 
 i ,n 
 
 k 
 
 Z 3* h , r(t .. , - t _,,) (36) 
 
 . _ i,n n-i n+l-i n+1 
 i=0 
 
 = -(-h n ) r , r = 1, 2,..., k+1 
 Define 
 
 Vl-i " t n+l 
 co . = 
 
 i h 
 
 h +h .+...+ h .. . 
 n n-1 n+l-i 
 
 From definition 1 
 
 and 
 
 A<_|oo.|<i, i >_ 1 (37) 
 
 = Uq 
 
 oj.-u.I > A if i 4 J (38) 
 
 Divide (36) by rh to get 
 
 , r 
 
 Z 3*. (w.-ok.. )uk = - — (39) 
 
 . _ i,n 1 l+l 1 r 
 i=0 
 
2k 
 
 The right hand side of (39) and the coefficients of s* on the left hand 
 
 i,n 
 
 side are bounded uniformly in n by (37) • Ike determinant D of the 
 coefficients is a simple multiple of the Vandermonde determinant, namely 
 
 D = [ n (w.-oj.)][ n (w.-o). ._)] 
 k>i>j>p X J i=0 ^ 1+1 
 
 By (38) this is hounded away from zero, hence the solution of (39) is 
 
 bounded uniformly in n . 
 
 Q.E.D. 
 
 Theorem 5 The variable step Adams -Bash forth (AB) 3 Adams -Bash forth- 
 
 Moulton predictor-corrector (ABM) 3 and Adams -Moulton (AM) methods are 
 stable and convergent for any step selection scheme. 
 
 {ho) 
 
 1 
 
 Proof The vector of saved values y is [y , hy',..., hy ' , . ] so 
 *-n n n n-k+1 
 
 the matrix A is given by 
 n 
 
 l,n 2,n k,n 
 
 6 n 6_ ... 6, 
 
 l,n 2,n k,n 
 
 while 2, is [0, 1, 0,. .., 0] for AB and [3* , 1, 0,. 
 — ti u ,n 
 
 and AM. 
 
 , 0] for ABM 
 
Hence, from equation (23), 
 
 S = (I -& eT) M A 
 n -n -2 n 
 
 25 
 
 1 
 
 n l,n 
 
 n 2,n 
 
 • • • 
 
 \,n 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 Ui) 
 
 where 
 
 i,n 
 
 3. for AB 
 i,n 
 
 i,n 
 
 for ABM and AM 
 
 From Theorem h, n are uniformly bounded, therefore, S is 
 i,n n 
 
 uniformly bounded. Hence the |S 
 
 n+mi 
 
 are uniformly bounded for 
 
 < m < k. From the structure of S , we see that for m > k 
 — — n 
 
 *n+m __ ~n+k 
 n n 
 
 1 x. 
 
 \ 
 
 (U2) 
 
 Hence the I Is are uniformly bounded for all m > n and the method 
 
 satisfies the stability condition. From Theorem k, A and I are 
 
 ' n -n 
 
 uniformly bounded. Finally, we can expand y(t . ) and hy' . n (t .) 
 
 n-i n-i-1 n-i 
 
 by Taylor's series about t with a remainder involving h y (?)• 
 
 If the order is greater than r-2 , then all but the remainder terms 
 
 r— 1 ( r ) i 
 
 will vanish, and we can bound d by h (h C Max y ). 
 
 1 '-n 1 ' n r '" ' 
 
 Thus, the three conditions of Theorem 2 are satisfied. 
 Q.E.D. 
 
26 
 
 Theorem 6 If the step selection scheme is such that at least k 
 
 steps of constant size are taken between step changes, then the k-step 
 interpolation AB 3 ABM 3 and AM methods are stable and convergent. 
 
 Proof Once again, we show that the hypothesis of Theorem 2 are 
 
 satisfied. We will use the Nordsieck vector to express the method as 
 
 a k+1 value method. Then the matrix S is given by 
 
 n 
 
 § - [I - A elf A C (1*3) 
 
 n z. n 
 
 = R C 
 n 
 
 The matrix R has the important properties that the simple eigenvalue one 
 
 corresponds to the right eigenvector e_ , and that all other eigenvalues 
 
 are zero (Gear [2], pp. 138-lUo). Consequently, R x = y e_ where 
 
 |y |/||x|| is uniformly bounded for all x (Tu [6], p. 20). The 
 
 eigenvalue one of C corresponds to the eigenvector e_ . When 
 
 h = h n , C =1. From the hypothesis of the theorem, it is not 
 n n-1 n ' 
 
 possible for C. ^ I ^ C. if |i-j| < k. Consequently, the product 
 
 S m = RC n RC . ...EC 
 n m-1 m-2 n 
 
 either contains less than 2k+2 matrices, or can be reduced to 
 
 S m = R C , . . . R C R k C , R^- k_n 
 n m-1 q q-k 
 
 where 
 
 <_ q-k-n < k 
 
 Since the ratio of adjacent steps is bounded by A , |C 
 
 ii m i i 
 
 is uniformly bounded. Hence, ||C R <_ k uniformly. Therefore, 
 
 if S contains 2k+2 or more matrices 
 n 
 
27 
 
 S m z=RC n ...RC R k C . R 1-k-n 
 n — m-1 q q-k n ± 
 
 = R C n ...EC R k x 
 m-1 q — 
 
 where | x | | <_ k^ | zj | . 
 Hence, 
 
 m 
 
 n — "m-1 ' ' " il "q '"x — i 
 
 S„ z = R C_ ., . . . R C_ y„ e 
 
 = y x% 
 
 so that 
 
 S m z|| = Iv 
 
 n — ' 
 
 x ' '— ' 
 
 1kg |x|| (as y,y||x|| is bounded] 
 
 l k 5 k 6 I 111 I 
 
 Therefore, ls m || < k k, 
 
 n 
 
 ^ ~6 
 
 If S contains less than 2k+2 matrices, S is bounded by the finite 
 
 n ' n 1 ' 
 
 product of the norms of each matrix, hence the method satisfies the 
 
 stability condition. 
 
 Theorem 3 shows that A and I are bounded, d is bounded 
 
 n -n -n 
 
 by the same argument as used in Theorem 5. Hence Theorem 2 proves 
 
 convergence and stability. 
 
 Q.E.D. 
 
 Tu [6] points out that the interpolation form of the Adams 
 formula is not necessarily stable under any step selection scheme. He 
 shows that for the three step ABM formula with 
 
 h 2i = Wh 2i+1 = h 2i + 2 = ^21+3 = * ' ' 
 
28 
 
 the matrix S has an eigenvalue of [(co-l) /Uoo] . For sufficiently 
 large w this is unbounded. Examples of instability when oj = 10 
 are given in that thesis along with examples showing that the variable 
 step technique is stable for the same step selection scheme. This case 
 points out that the variable step technique is more stable in some cases. 
 
 Another example in Tu's report substantiates Brayton, et.al.'s 
 claim that the variable step technique is more stable. The problem 
 y' = -y with h = .05 and h = .005 was integrated using a three 
 step backward differentiation formula. It was apparently stable for the 
 variable step technique and apparently unstable for the interpolation 
 technique. It is hypothesized that the variable step method can also 
 cause instabilities, but no examples have yet been found. 
 
 In practice, we do not vary the step in an extreme way. When 
 the step changes are controlled to be small, both techniques are stable 
 as shown below. 
 
 Theorem 7 If a method satisfies the stability condition for fixed 
 
 stepSj it also does so with respect to a step selection scheme that 
 produces step size changes small in the sense that 
 
 h n+l 
 
 = 1 + 0(h) as h + (44) 
 
 h 
 n 
 
 Note 
 
 If 6(t, h) = 6(t) and is a boundedly differentiable function 
 
 of t, then the hypotheses of the theorem are satisfied since 
 
 h ._ 6(t .,) 6'(£ ) 
 
 n+1 n+1 _ , . . n 
 
 "h~ = e(t ) ~ 1 + \ "em 
 
 n n n 
 
 = 1 + h0'(C ) 
 
 ii 
 
 = 1 + 0(h) 
 

 29 
 
 Proof We will use the hypotheses of the theorem to decompose S 
 into the form 
 
 S = S + h S (U5) 
 
 n n n 
 
 where S is the form of S when constant steps are used and S can be 
 
 n n 
 
 bounded. (This step will be delayed until Lemma 3 below.) Then we 
 can examine I Is II as follows. For fixed n, set 
 
 x=Sx m>n (U6) 
 
 -m n — n — 
 
 Then 
 
 x ... = S x 
 -m+1 m -m 
 
 = Sx+hSx (1+7) 
 
 -m m m -m 
 
 If the constant step method satisfies the stability condition, then 
 
 Is "l| < k rt . Lemma 3 will show that Is < k, . Therefore, Lemma 2 
 ii ii_q ' ' n M — • 1 
 
 can be applied to (^7) with k_ = . Hence, 
 
 k k ,T 
 
 M M I io m ll^i 01 II II 
 
 x =S x <k^e x 
 i i — m i i M n -n" - ''-n 11 
 
 Hence, 
 
 k k T 
 l|s m || :k n e 01 
 
 I I n l I _ Q 
 
 so the method satisfies the stability condition. 
 Q.E.D. 
 
 Lemma 3 If a step selection scheme satisfies the hypothesis of 
 
 Theorem 7, S can be written as S + h S where S is independent of 
 n n n * 
 
 h and n, and j |s is uniformly bounded. 
 
30 
 
 Proof 
 
 Part I : Interpolation Technique 
 From (9) and (23) 
 
 T M 
 S = (I - SL e_) AC 
 
 n -n -2 n 
 
 where SL is independent of n. Now 
 
 C n = diag(l, h n /h n _ 15 (h n /h n ^) 2 9 ...) 
 
 =I + diag(0, \\_ ± -l, (h^h^) 2 -!,...) 
 
 = I + 0(h) 
 = I + 0(h ) 
 
 since h /h = 1 + 0(h) and h < h A -1 
 ii n— jl — n 
 
 Therefore , 
 
 Hence, 
 
 S = (I - I e^) M A(I + 0(h )) = S[I + 0(h )] 
 n 2 n n 
 
 S = S 0(h )/h 
 n n n 
 
 is uniformly bounded. 
 
 Part II : Variable Step Technique 
 
 It is shown in Tu [6], p. 52, that the coefficients a and 
 
 have the form 
 
 a. = a. + R (v ,. . ., v ) 
 
 l ,n l a. n n-k+2 
 
 l 
 
 3 n- n = S n + R ft (V '•'•' V r, V + 9 ^ 
 
 i ,n i p. n n-K+£ 
 
 where 
 
 h. - h. , 
 v = x J-" 1 
 
 1 h i-l 
 
31 
 
 where the R's are rational functions that vanish at v = v _ =...= \) , ,_ - 
 
 n n-1 n-k+2 
 
 (when the step sizes are equal), and where the a. and 6. are the corresponding 
 
 fixed step coefficients. (This result follows directly from the nonsingular 
 
 system of equations that determine a and £.) From the hypothesis of 
 
 Theorem 7, v . = 0(h) , hence R = 0(h) = 0(h ). Therefore, we can express S , 
 
 which involves the coefficients a, 3, a* and 3* 5 as S + h S , where S is the 
 
 n n 
 
 matrix obtained by setting the a. equal to a. , etc., and S includes the 
 
 l ,n l n 
 
 terms R divided by h. These can be uniformly bounded. 
 Q.E.D. 
 
32 
 
 PRACTICAL IMPLICATIONS 
 
 It can be pointed out that a common step selection scheme 
 
 satisfies the hypotheses of Theorem 7 • If the step h is controlled 
 
 so that the local error estimate is equal to e(or eh ), and if the truncation 
 
 n 
 
 r+1 r+2 
 
 error estimate is <J>(t ) h + 0(h ) where <b(t) is the principle error 
 
 r n n n r ^ * 
 
 function for the method, we have 
 
 <|>(t ) h r+1 + 0(h r+2 ) = e (or eh ) 
 n n n n 
 
 Hence , 
 
 h. = ( -tt| J q + 0(h n ) where q = r+1 or r, 
 n 
 
 provided that <j>(t ) is not zero. If <\>(t) is bounded away from zero and 
 
 <J>'(t) is bounded above, then 
 
 n n+1 
 
 = 1 + 0(h ) 
 n 
 
 and h . /h > A > 0. Consequently, we can expect either method to be 
 mm max — 
 
 stable if the fixed step method is stable. 
 
 However, in practice we do not send h to the limit and we often 
 
 do not bother to adjust the step if the change is small. Therefore, there 
 
 is reason to prefer the variable step approach. Brayton, et.al. [l] point 
 
 out that the amount of work in evaluating A y is less in the variable step 
 
 approach than in the interpolation/Nordsieck approach, but that the 
 
 coefficients in A and i must be recomputed for each step. For a large 
 n -n 
 
 system of equations, this is not important as A is the same for each 
 equation in the system. For a few equations this is a serious consideration, 
 which leads us to recommend variable step methods for large systems of 
 equations and interpolation methods with the Nordsieck vector for small systems 
 
LIST OF REFERENCES 
 
 33 
 
 
 [l] Brayton, R. K., Gustavson, F. G., and Hachtel, R. D. "A New 
 Efficient Algorithm for Solving Differential -Algebraic 
 Systems Using Implicit Backward Difference Formulas," 
 Proceedings of the IEEE , 6p_, #1, pp. 98-108, 1972. 
 
 ]2] Gear, C. W. NUMERICAL INITIAL VALUE PROBLEMS IN ORDINARY 
 
 DIFFERENTIAL EQUATIONS, Prentice -Hall, Inc., New Jersey, 1971, 
 
 [3] Henrici, P. DISCRETE VARIABLE METHODS FOR ORDINARY 
 DIFFERENTIAL EQUATIONS, Wiley, New York, 1962. 
 
 [k] Nordsieck, A. "On the Numerical Integration of Ordinary 
 
 Differential Equations," Math . Comp . , l6, pp. 22-^9, 1962 . 
 
 [5] Piotrowski, P. "Stability, Consistency and Convergence of 
 Variable K-Step Methods," Conference on the Numerical 
 Solution of Differential Equations , 109 , Springer, Berlin, 1969 
 
 [6] Tu, K. W. "Stability and Convergence of General Multistep 
 
 and Multivalue Methods with Variable Step Size," Department 
 of Computer Science Report $26 , University of Illinois at 
 Urban a- Champaign, July 1972 (Ph.D. Thesis). 
 
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 2. TITLE 
 
 THE EFFECT OF VAEIABLE MESH SIZE ON 
 THE STABILITY OF MULTISTEP METHODS 
 
 3. TYPE OF DOCUMENT (Check one): 
 
 Pel a. Scientific and technical report 
 
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BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-73-570 
 
 4. Title and Subtitle 
 
 THE EFFECT OF VARIABLE MESH SIZE ON 
 THE STABILITY OF MULTISTEP METHODS 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 April 1973 
 
 7. Author(s) 
 
 C. W. Gear, K. W. Tu 
 
 8. Performing Organization Rept. 
 
 No. 
 
 9. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urban a, Illinois 6l801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 US AEC AT(ll-l)lU69 
 
 12. Sponsoring Organization Name and Address 
 
 US AEC Chicago Operations Office 
 9800 South Cass Avenue 
 Argonne, Illinois 60^39 
 
 13. Type of Report & Period 
 Covered 
 
 technical 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 The effects of two different techniques for implemementing 
 variable mesh sizes in multistep formulas are investigated. It is 
 proved that one is more stable than the other for some cases , but that 
 both are stable when the step changes are small. The practical impli- 
 cations of these results are discussed. 
 
 17. Key Words and Document Analysis. 17o. Descriptors 
 
 interpolation 
 variable step 
 
 17b. Identifiers/Open-Ended Terms 
 
 17c. COSATI I- ic Id /Group 
 
 18. Availability Statement 
 
 unlimited 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 curitv Class (Thi 
 
 20. Security Class (This 
 
 Page 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 35 
 
 22. Price 
 
 FORM NTIS-38 (10-70) 
 
 USCOMM-DC 40329-P7I 
 

OCT 2 9 1973 
 
nUOHVlKIHul