maw,! 1 B iiilBBi SffK IMS BH i3ffiBBH8B Mt^BiIUHflWIIalfifllnlll HBotlfimiflll HHMhui ■UVwHDh kkjo lunn MnflB HjHB9 ffi rem nHHa im mom 1 ImBeU B2HK B HBH BHUI BtSfl ■ Nnj »«i BiBHfifil H Mm BMMMMMH mi In b§I In RflBB tJ$M Hi Bhheh tan JH Wt 1 1 HwHmlfHnmmw m S ■ fl KgUIHHB ■H «sSl mimiiM HnfllBBIHllffw U JEw Ira Bra B Hi LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN cop- *» Digitized by the Internet Archive in 2013 http://archive.org/details/stabilityconverg526tuka £/L UIUCDCS-R-72-526 '^yiaZL STABILITY AND CONVERGENCE OF GENERAL MULTISTEP AND MULTIVALUE METHODS WITH VARIABLE STEP SIZE KAI-WEN TU July 1972 UIUCDCS-R-72-526 STABILITY AND CONVERGENCE OF GENERAL MULTISTEP AND MULTI VALUE METHODS WITH VARIABLE STEP SIZE* KAI-WEN TU July 1972 DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS URBANA, ILLINOIS 6l801 This paper was advised "by Dr. C. W. Gear, Department of Computer Science, for K. W. Tu, Department of Mathematics, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics. Ill ACKNOWLEDGMENT I wish to express my sincere gratitude to ray thesis advisor, Professor C. William Gear, for many invaluable suggestions and criticisms which motivated and contributed greatly to this work. Computing time was also provided by Professor Gear. Thanks are due to Mrs. Barbara Armstrong for the efficient secretarial help, particularly the typing of the entire manuscript. This work is dedicated to my parents. IV TABLE OF CONTENTS Page 1. INTRODUCTION: VARIABLE MULTISTEP AND MULTIVALUE METHODS. ... 1 1.1 Interpolation Method Using Nordsieck Vector k 1.2 Variable Step Method 7 1 . 3 Comparison of the Two Methods 12 2. STABILITY AND CONVERGENCE OF ADAMS METHODS lk 2.1 Interpolation Method 1)4 2.2 Variable Step Method 32 3. STABILITY AND CONVERGENCE OF GENERAL MULTISTEP METHODS kk 3-1 Interpolation Method kk 3.2 Variable Step Method 50 3.3 Numerical Testing of Stiff Interpolation Method and Stiff Variable Step Method 60 k. PRACTICAL CONSIDERATIONS OF THE STABILITY AND CONVERGENCE THEOREMS 69 LIST OF REFERENCES 71 APPENDICES A. PROGRAM FOR THE INTEGRATION OF y» = -y , y(0) = 1 USING 1+-VALUE ADAMS INTERPOLATION METHOD WITH STEP SIZE BEING ALTERNATED ONCE EVERY THREE STEPS. . . 72 B. PROGRAM FOR THE INTEGRATION OF y' = -y, y(0) = 1 USING 3 -STEP ADAMS VARIABLE STEP METHOD WITH STEP SIZE BEING ALTERNATED CONSTANTLY 75 Page PROGRAM FOR THE INTEGRATION OF y' = -y, y(0) = 1 USING U-VALUE STIFF INTERPOLATION METHOD WITH STEP SIZE BEING ALTERNATED CONSTANTLY 78 PROGRAM FOR THE INTEGRATION OF y' = -y, y(O) = 1 USING 3 -STEP STIFF VARIABLE STEP METHOD WITH STEP SIZE BEING ALTERNATED CONSTANTLY 8l 1. INTRODUCTION: VARIABLE MULTISTEP AND MULTIVALUE METHODS Consider the initial value problem for the single differential equation y' = f(y,t), y(0) = y Q (l.l) We assume that f(y,t) satisfies appropriate Lipschitz and continuity conditions such that a unique solution y(t) exists and y(t) e C'[o,b]. Later on it is assumed that y(t) has as many higher derivatives as necessary. There are various methods to compute the numerical approximation of y(t). Among them are the general k-step methods in the form k I (a. y . + hg. f .) = (1.2) . - l n-i i n-i i=0 where h (a constant) E step size y . E the numerical approximation of y(t) at t . e (n-i)h n-i n-i f .. E f(y . , t .) n-i n-i n-i a. and B. are constants, l l It has long been established that by specifying an appropriate set of values for a. and 8. one can obtain a stable and convergent k-step 11 method. Moreover, there exists more than one k-step method. For instance, Adams-Bashforth methods (see Henrici [9], Section 5.1), Adams -Moul ton methods (see Henrici [9], Section 5.1), and stiffly stable methods (see Gear [8], Section 11. l) are all special cases of (1.2). In this thesis we are concerned with variable step size mutlistep methods which are generalizations of (1.2). It turns out that a. and B. are no longer constants. Even though they are still independent of y . and f . , they are dependent on the step sizes taken. We are also going to consider multivalue methods (see Gear [8], Section T»l) which in the case of constant step size h have the following form 4,(0) :: B V-i (1.3) Y . .» • Y M + C G(Y / %) m - 0. 1,...(1.M -n,(m+l) -n,(m) — -n,(m) • * ••*■'■• ' where B is a constant matrix, C a constant column vector, Y , is the — -n-1 column vector of saved information iT ln_! » ty n . r y n- 2»" y n .k» hy n-r - hy n-k ] and G(Y ( N ) = h(f(y f ,, t ) - y' , J -n,(m) w n,(m)' n n,(m) (The description of this procedure (l.3)-(l.5) will be elaborated in Section 1.1. ) Predictor-corrector multistep methods are a particular case of multivalue methods. For instance, a U-value Adams -Bashforth-Moul ton method has the form (1.5) B = 1 23/12 3 1 -16/12 5/12 -3 1 1 C r 3/8 1 When step changing is necessary, we shall consider an equivalent representation of (1.3) and (l.U) in an attempt to make step changing simple. This is based on the use of the Nordsieck vector [ll] a -n .1= [ Vr'Ci hk ^l /k!lT instead of Y . Accordingly, we have the representation a , v = A a -n,(0) -n-1 (1.6) a / , . v = a / x + I F(a , J (1.7) -n,(m+l) -n,(m) — -n,(m) equivalent to (2.3) and (2.U). Here, again, A is a constant matrix, (2) (k) I a constant vector, F the same function as G, and y' n , y ,v ' v - " ' J n-1' J n-1 »•*•» ^n-i are numerical derivatives of y _ solely used in the multi value method. n-1 Note that for instance y 1 % f(y _ , t _ ) . It was shown by Gear [8] that a general multistep predictor-corrector method can he expressed by representation (1.6) -(1.7). To demonstrate the correspondence between (l.3)-(lA) and (l.6)-(l.7) let us consider U-value Adams -Bashforth-Moulton method. In T this case, Y . is [y _, hy ! . , hy' ... hy' _] . The information stored -n-1 n-1 n-1 n-2 n-3 in Y , and a , determines the same third degree interpolating poly- -n-1 -n-1 nomial [ 8 ]• Therefore, Y . and a . are related by a transformati —n-1 -n-1 T, i.e. on a = T Y . -n-1 -n-1 where T = 1 1 3A -1 iA i/6 -1/3 1/6 Consequently, we have for the representation (1.6) -(1.7) A = TBT ,-1 1111 12 3 13 1 3/8 1 I = TC = 3A 1/6 Thus, our approach to step changing is two -fold. On the one hand we look at the interpolation method "based on multivalue methods in Nordsieck form, and on the other we consider a variable step method based on (1.2). We shall elaborate on these two approaches here. 1.1 Interpolation Method Using Nordsieck Vector Since we are working with variable step size, we shall assume throughout this paper that h is the size of step advancing from time t . to t , i.e. h -,=t -t _ . We adopt the conventional use of y n-1 n n-1 n n-1 n-i to indicate the numerical approximation of the solution y(t) at t . and n-i naturally f . = f (y . , t . ) . n-i n-i n-i Step changing for a Nordsieck vector a is easily accomplished by the multiplication by the matrix 'n-1 ■ o n-1 n-1 when u n-1 Formally , is called a step changing factor, n-2 (1.8) -n-1 h y n-l n-1 y n-l • • = k • n-1 (JO k! ''n-l n-1 o o n-1 n-1 h n y' n-2 °n-l h n-2 (k) k! y n-l n-l-n-1 (1.9) We define the step changing algorithm from t . to t as follows n-1 n 1. Prescribe the step changing factor oo n-1 2. Use (1.8) to obtain C n-1' 3. Use (1.9) to find the Nordsieck vector a* . —n-1 with step size h n-1' h. Replace a , in (1.6) by a* , . -n-1 —n-1 Incorporating this algorithm into (l.6)-(l.7) we obtain ^i,(0) " A C n-1 Vl a ,_ N + I F(a_ / _^) -n,(m+l) — n,(m) - 1V2 Ti,(m) This modified representation (l.lO)-(l.ll ) is called an interpolation method since the step changing involves an interpolation process. The prediction process (1.10) does not contain any knowledge of the differential equation. We just extrapolate a ,.x from a _, — n,(,U; -n-1 we need to correct the amount by which a , nS does not satisfy the -n,(0) (1.10) (1.11) So differential equation; this is performed in (l.ll). Theoretically, one may repeat the correction as many times as one pleases to attain convergence. However, correction steps call for additional computation and about two such steps are usually recommended. Suppose a , s is the value at the final correction step, then a x a ,,,>.. *' -n -n,(M) Suppose the matrix A is a Pascal triangular matrix whose (i,j) element is ( . ) , k > j ■•_ i >_ 0, and the step size is kept constant through- out the numerical integration, then there exists k-step methods of maximum order that are stable. (For constant step method, the meaning of order and stability are in accordance with Gear [8] or Henrici [9])- A proof of this is given in Gear [8]. What we intend to pursue here is the case when the step size is allowed to vary. We examine whether this will affect the stability and convergence of the method. We found that with "mildly" changing step size, the interpolation method remains stable and convergent. A detailed proof and discussion is given in Chapter 3. We substantiate the description of interpolation method with an example . Example 1.1 Three-step Adams-Bashforth predictor and Adams -Moulton corrector expressed in the form of interpolation method . y n,(0) h n-i y n,(0) n-1 n 2 y n,(0) n-1 1111 12 3 13 1 T" y n,(0) y n,(m+l) h n-l y n,(m+l) n-1 „ °'& '& 2 °n,(m+l) n-1 6 y n,(m+l) y n,(m) n-1 n,(m) h 2 y n,(m) 6 y n,(m) ( S 3 n-1 h v 1 n-2 *n-l n-2 „ 2 "n-1 n-2 T" y n-l J 3/8 1 3/U 1/8 h n-l (f(y n,(m)' t n ) " y n,(m) ) m = 0, l s . . . 1.2 Variable Step Method Consider the 3-step Adams-Bash forth predictor with constant step size h y = y + ^ (23 f - 16 f +5f ,) (1.12) n n-1 12 n-1 n-2 n-3 One way to derive (1.12) is to require that the method be exact for all polynomials of degree less than or equal to three. We shall carry this technique over to the variable step case to compute the coefficients & , . § , . § n and B -, in the equation n,l n,l n,2 n,3 y = a y + h .(§ . f - + f + B _ f ,) n n,l n-1 n-1 n,l n-1 n,2 n-2 n,3 n-3 (1.13) 2 "3 Consider the polynomials 1, (t-t _), (t - t _) and (t - t ) , n-3 n-3 n-3 The requirement for them to be exact in (1.13) yields a system of four linear equations r n,l = 1 (l.iiO (t n -t ) d + h 3 +h fl + h 6 , - (t -t ) n-1 n-3 n,l n-1 n,l n-1 n,2 n-1 n,3 n n-3 (t n-l " W 2 *n.l + 2 h n-l ( Vl ' W g n,l + 2 h^ (t n _ 2 - t n _ 3 ) 6 n>2 = (t n - t n _ 3 ) 2 (t - t .) 3 a + 3 h (t . - t _) 2 g . n-1 n-3 n,l n-1 n-1 n-3 n,l + 3 h ft - t „) 2 B = (t - t J 3 L n-1 n-2 n-3 n,2 n n-3 Upon solving (l.lU) we have r a = 1 n,l 3 , = 1 + n,l (1.15) h n-l (2 h n-l + 6 h n-2 + 3 h n - 3 ) 6 h n-2 (h n-2 + h n -3 } n,2 h n (2h +3h G +3h .) n-1 n-1 n-2 n-3 '~ 6 h h n-2 n-3 h (2 h + 3 h ) n-1 n-1 n-2 P n,3 6 h (h + h ~) n-3 n-2 n-3 This illustrates the derivation of variable step methods for 3-step Adams -Bashforth predictor methods. Motivated by the preceding example, we proceed to construct the variable step method for a general multistep method. For each k-step method (1.2) we define an equation E (d . y . + h i=0 n,i n-i n-1 n,i n-i f . ) = (1.16) where a . and $ . are functions of the step sizes h , . h ^,..., n,i n,i n-1 n-2 h , , as the corresponding variable step method . We require that 1. a. = => d . = 0, 3. = =>§ . = (1.17) i n,i i n,i 2. For some non-vanishing a.'s and 3.'s we specify the corresponding a . 's and 3 . 's such that n,i n,i a . = a. + 0(h _), 3 . = 6. + 0(h _) (l.l8) n,i i n-1 n,i i n-1 3. For a r-th order method the total number of prescribed & . 's and 3 .'s, vanishing or non -vanishing, should n,i n ,i be 2k + l - r. h. The polynomials 1, (t-t ),..., (t-t ) satisfy n — k. n "tK. (l.l6) exactly. It is worthwhile to explain the purpose of the above conditions l.-k. First let us consider a linear difference operator L given below. n k L [y(t )] = Z (d , y(t .) + h 8 . f(y(t .), t .)) (1.19) n n .__ n,i n-i n-1 n,i n-i n-i Without loss of generality one may assume d . = -1. If we substitute n,0 the Taylor series expansion of y(t .) and y'(t .) = f(y(t . ), t .) n-i n-i n-i n-i about t , for i = 1, . . . , k and factor out the terms y(t , ) , y * (t ,),... in (1.19) , we get L n [ ^ (t n )] - C „0 ^n-k' + °nl \-l *' + ° ^ and this linear operator L is said to be of order r. Equation (1.22) can be accomplished in a different, yet more convenient manner by condition k, Now (1.22) yields a system of (r+l) linear equations with (2k+l) unknowns a , ..., a , 3 ,...,6 . Dahlquist [7] pointed out that n , J- n yK n , U n ,K in the case of constant step k-step methods the maximum order for stability is k+1 if k is odd k+2 if k is even This means r <_ k+1 if k is odd (1.210 r <_ k+2 if k is even The specifications 1.-3. allow us to arrive at a system of (r+l) non- homogeneous linear equations with r satisfying (1.24). For such a system unique solutions of the unknown & . 's and 3 . 's exist. Their evaluation n,i n,i constitutes the major task of variable step methods. Therefore, we shall state here an algorithm to determine all coefficients a .'s and 3 • 's. n,i n,i 11 1. Impose conditions 1.-3. 2. Use condition k. to obtain a system of (r+l) linear equations with (r+l) unknowns d . 's and 3 .'s. n,i n,i 3. Solve for the unspecified d . 's and (3 .'s. n,i n,i We will prove that a variable step method is also stable for "mildly" varying step sizes. For Adams method the stability proof of this method requires much less restrictive conditions than interpolation methods. This advantage will be demonstrated in the numerical test in Example 2.3. A detailed treatment of the general proof of stability and convergence will be offered in Chapter 3. For a complete illustration of predictor-corrector methods in the form of variable step methods, consider . Example 1.2 Three-step Adams -Bashforth predictor and Adams -Moult on corrector method . A-B Predictor y , rtl = y , + h_ , {&] f_ , + S. (p > f_ n + &l f_ J where n,(0) " °n-l n-1 n,l n-1 n,2 n-2 n,3 n-3 /x h.(2h n +6h_+3h_) i(p) _ -, + n-1 n-1 ^2 n-3 n,l 6 h (h + h ) n-2 n-2 n-3 / x h (2 h . + 3 h + 3 h ) ,IP) _ n-1 n-1 n-2 n-3 n,2 ' 6 h _ h ~ n-2 n-3 r \ h . (2 h . + 3 h _) (p) .. n-1 n-1 n-2 n,3 " Th (h _ + h .) n-3 . n-2 nr3 12 A-M Corrector y , , = y + h (S (c) f , % + 6 (c) f + 3 (c) f + B (c) f ) y n,(m+l) y n-l n-1 n,0 n,(ni) n,l n-1 n,2 n-2 P n,3 n-3 where (n) , h . (h _ + U h + 2 h J h 2 1 (h + 2 h + 2 h J 'Ui _ 1 n-1 n-1 n-2 n-3 n-1 n-1 n-2 n-3 n,0 "" 2 ' 12 h (h + h _) ' 12 h h , (h + h ) n-2 n-2 n-3 n-2 n-3 n-1 n-2 h 2 , (h +2h ) n-1 n-1 n-2 12 h _ (h " + h )(h + h _ + h _) n-3 n-2 n-3 n-1 n-2 n-3 „/ n n h . (h + U h Q + 2 h _) "(c) 1 + n-1 n-1 n-2 n-3 n,l 2 12 h " (h " + h .) ' n-2 n-2 n-3 t \ h 2 .(h,+2h +2h ) -(c) _ n-1 n-1 n-2 n-3 n,2 " 12 h h (h . + h ■ ) n-2 n-3 n-1 n-2 ( \ h 2 n (h + 2 h _) "(c) _ n-1 n-1 n-2 n,3 == 12 h TTh " + h )(h + h " + h Z) n-3 n-2 n-3 n-1 n-2 n-3 f , x = f(y , v, t ) m = 0, 1,. . . n,(m) w n,(m) n 1.3 Comparison of the Two Methods Both interpolation methods and variable step methods are derived from (1.2) and they reduce to (1.2) when the step size is held constant throughout the numerical computation. With variable step methods one computes y directly from the k preceding points and their derivatives, whereas for interpolation methods, one computes y from the Nordsieck vector at t . When we represent the variable step method in the Nordsieck form, we obtain a modified version of the interpolation method with the column vector l_ being a constant vector. Brayton [3] has shown in the stiff case that these two methods are not equivalent when the step size varies. 13 It has been definitely established in Chapter 2 that corresponding to Adams's scheme the variable step method is more stable than the interpolation method. For the stiffly stable methods the same claim is supported by numerical testings only. However, it will be shown in Chapter 3 that a more restrictive condition, namely, that the ratio of successive step sizes satisfies V^- ^- (1 - 25) n-1 is sufficient for either step changing method to be stable if the corresponding fixed step method is stable. Ik 2. STABILITY AND CONVERGENCE OF ADAMS METHODS In this chapter we are going to examine in detail the interpolation method and variable step method based on Adams -Bash forth predictor and Adams-Moulton corrector (equivalent names: Adams method or Adams-Bashforth-Moulton method) methods. The convergence behavior of either method will be analyzed and will be exhibited in examples. 2.1 Interpolation Method As introduced in Chapter 1, the representation of the interpolation method is given by (l.lO)-(l.ll ) Sn,(0) = A °n-l Vl a i ^ \ = a ( \ + l F ( a i \ )• — n,(m+l) -n,(mj — -n,(m) With an appropriate l_ this represents an Adams-Bashforth predictor and Adams-Moulton corrector. Let a(t ) be the correct value of a at t = t . Consi^ — n — n S , n v = A C a(t ) (2.1) -n,(0J n-1 — n-1 a , . = a , x + I F(a f v (2.2) -n,(m+l) -n,(m) — -n,(m) Then the local truncation error d is defined as — n d = a - a(t ) (2.3) n -n — n Let e / v = a / > - a, / x (2.U) -n,(m) -n,(mj -n,(m) e = a - a(t ) (2.5) n -n n Subtracting (2.1) from (1.10) and (2.2) from (l.ll) we have (with the notations given in (2.3), (2.U) and (2.5)), 15 e fn s ■» A C . e (2.6) -n,(0; n-1 n-1 ' e -n , .^e . + Jl (F(a , J - F(a , J) (2.7) ,,(m+l) - n ,(m) - n,(m)' v rn,(m)" By the mean value theorem F(a , J - F(a , J = ~ (E , J e , v (2.8) n,(m)' -n,(m) 3a '^(m) -n,(m) where - — ( £ / v ) is a row vector evaluated at a point between 3a_ ■ 2 n,(m) * a / x and a / \. Substitute (2.8) in (2.7) to get -n,(m) -n,(m) e ( J .,\ = (! + £ ¥~ (I f J) e / \ (2-9) -n,(m+l) — 3a_ -n,(m) -n,(m) Without loss of generality, one may assume that there are always M corrector iterations. Combine (2.6) and (2.9) M-l e n,(M) \l Q < I+ i!i (£„,(!)» ",-xVl (2 - 10) Let Then M_1 3f S = n (I + *.-g- (E ,,J) A (2.11) n i=0 3^ -n,(i) e / M x = S C e (2.12) -n,(M) n n-1 -n-1 The local error e given by (2.5) is rearranged in a computable form as follows % = S* - a ^ n ) (2.13) = ^,(M) " a(t n } = 2^,(M) " ^i.(M) + ^n,(M) " a(t n ) -n,(M) -n Substituting (2.12) into (2.13) 16 e = S C . e _ + d -n n n-1 -n-1 -n 9F For Adams methods — (£ / . \ ) has the following form (2.1U) 3F ~ (£ ,,J = [h . f , -1, 0,..., 0]k 3a ^rijCi) n-1 y ' ? — — n (i) = h n -iV((i)^ T - 5 -i (2.15) where £ / . \ is the 0-th component (all numbering starts from 0) of n ? \ i ) £ / . n and T 6. = —l 1+1 Thus, (2.11) becomes M-l (2.16) M-l Suppose f is bounded, i.e. there exists a constant L > such that If I < L I y| _ (2. IT) Then it is clear that upon expanding the right side of (2.l6) we have S = S + h n S n n-1 n (2.18) where s = (i -L^ M A (2.19) and S is a matrix whose elements are polynomials in h . with bounded n " n-1 coefficients. From (2.lU) and (2.18) IT e =(S+h n S)C n e + d (2.20) -n n-1 n n-1 -n-1 -n n n * I n (S C. ,) (h, . S, C, _ e 4 . + d.) + n \ (s 1 such that s ll sm |ll k for all integer m ^ 1 (2.2*0 Let us demonstrate the above by considering a U-value Adams-Bashforth- Moulton method. From (2.19) 18 ■( 10 3/8 10 1 1 3A 1 1/6 [0, 1, 0, Q]\ M 1 1 1 1 1 2 3 1 3 1 1 3/8 M 3/k 1 1/6 o 1 1 1 1 1 1 2 3 1 3 1 s 2 = 1 -3/8 -3/1+ 1 -1/6 1 1 5/8 iA -3A -1/2 -1/6 -1/3 1 11/2U 1/6 1/1+ 2/3 1 1 1 — 1 1 2 3 1 3 1 for any integer M >_ 1 3A 1/2 19 S 3 = 1 1/2 1/6 s" = 1 1/2 1/6 The fact that the second, third, and fourth row of S J vanish is highly- desirable. Consider 2 U). 03. l 1 1 2 6 s 3 c. = 1 Thus, 3 3 3 S C. S" 3 C... .S° C = i J m 2 1 f f ° (2.25) 20 It is clear that C , is an identity matrix when there is no step size n-1 r change from t to t . Suppose the step size does not change more than every third step and that C i- I, then (2.21) yields for n-j ^3 J n n i=j+l < S be a constant such that < W. (Note that it is min h natural to assume > 1, •. W > l) . Suppose the step size does not min change more than every k consecutive steps. Then N n (sc. J 1=J+1 is uniformly bounded for all j = 0, 1,..., N-l. 22 Proof First we observe that for an arbitrary step changing matrix C. . Hence, as long as N k n (S C ) contains a factor S , (2.26) and (2. 27) lead to 1=3+1 X N N || n (s c )| | = sup || n (s c ) x| | i=j+l 11x11=1 i=J + l X < k — s Let the J be the index set N {The values of j such that n (S C._ ) fails i=j+l X k to contain a factor S } From the hypothesis, J contains less than 2k-l elements. Thus, we see that N Max || n (S C. _)|| i (k ^f k ~ 2 v 2k— 2 Let k = (k w ) . Summing up the above two cases, we have J_ s N II n (S C. )|| <.k if. 3 (2.29) i=0+l The bound k depends neither on N nor on j . Q.E.D. Since we are dealing with methods of at least second order, the local truncation error d. satisfies i 23 ijl < h._ x e ¥i (2.30) where e is 0(h. _). It is also clear that there exists l-l a constant k > such that ||S C || < k ?j (2.31) Having established bounds for various terms in (2.20), we are in a good position to present the convergence theorem. Theorem 2.1 The interpolation method based on (k+l) -value Adams-Bashforth- Moulton method for the first order equation (l.l) is convergent under step size changing conditions provided the change is not made more frequently than every k steps. Proof At time t„ the error e, T satisfies N N Let Then e, T = Z n (S C. . )(h. _ S. C. . e. + d.) -N J=1 i=( . +1 i-l J-l J J-l -J-l -J N + n (S C. .) e. . . i-l —o 1=1 E x = |e I + ^- (2.32) n ' ■ n ■ ' k 2 Sllil-IA ." (s c i-i )(h j-i § j ViVi + ij )|l + H" (sc i-i'Sol J _L -L J J- N -^ k l k 2 h j-l X j-1 + k l l|e o'l (2 - 33> 2k by applying (2.29), (2.30) and (2.31). Consequently, *i * k i k 2 h j-i x j-i + S N-oll + ir- < 2 -3 u > We shall show "by mathematical induction that k k (t„-t ) ^i^ll-oll*^)- 1 2 " ° (2-35) 1. When N = 0, consider (2.32) <^II« II ♦ £ since k >_ 1. 2. Assume (2.35) holds for N £n From (2.3*0 n+1 x j.1 : z k k. h. x. . +L i|eji +f n+1 — 1 2 j-1 j-1 1 ' ' 0' ' k2 n+1 p k. k (t. -t_) j=l d + k, lie' II + .— 1 M M kg n+1 k k (t. -t ) = ( kl ||e || ♦ £) 0, le^l | ■*■ for all N such that t. T e [0,b]. This means the max ' ' N 1 ' N computed solution converges to the exact solution as h ■+ max Q.E.D. In the preceding proof the step size change is restricted to every k steps in order that n (S C. ) based on a general (k+l) -value i Adams method would be uniformly bounded. We like to point out that this restriction could be removed in the case of the 3-value Adams method 26 I = 5/12 1 1/2 S = 10 5/12 10 - 1 1 1/2 jo, i, oJ — 1 1 1 1 2 1 Consequently, 1 7/12 1/6 -1/2 for any integer M > 1 1 7u J 12 2 ID... 03. _ J +1 J 12 1 2 N n (s c ) = i=j+l Since o> . < W ¥■ j, the above matrix is uniformly bounded. Therefore, we J can obtain convergence without being restricted on the frequency of step size change. However, the situation is different for higher value Adams method, If the step size change is subject to gross changes, then the error may grow rapidly throughout the numerical computation. Consider 27 Example 2.1 The error growth of the U-value Adams method with, alternating step size . In this example we are mainly concerned with the behavior of N n (S C ). It is easy to see that II (S C ) can be converted to i 1_ i-j+1 the matrix 1 X X X _ . . TO .N-l _ N-l M ... , . _ , _ N-l N-l . / _vN-j,lvN-j 2 , s f xN-j+l,3wlxN-j-l 2 _ , » o x (-1) °(— ) n w. n (l-w ) (-D (r)(p) to n u. n (i-a>.) 1=0 i=j+l °i=j i=j+l o x (.dN-^^^-mV^ Yu-uO (-i)^ +1 (^ u .Y^Y(i- u .) J * i=j 1 i=j+l x J i=j i=j+l x where x denotes other non-zero elements in the matrix. The eigenvalues of this matrix are determined to be „ . N-l £ (l-w. ) i, o, o, (-i) N " J n x 2 x i=j Of the above eigenvalues, only the last one is affected by step size change. Suppose for any two consecutive steps the product of their step size changing factor is unity. Then for some oj > 1 -1 w 2j " w 2j+l ~ w When N-j is even 28 2 H , N-l w- (.1-0) ) N-l 1-w (-D N " J n X § * 1 - I n (-ylll (2.37) 4o) when N-j is odd _ . N-l ok (1-oa.) ,_ , N-2 o) 2 (l-o).) |(-D N - J n JL_^.pIl=al n i 1 1 (2.38) i*J i= J . u,(a,-l) r (o)-l) 2 1 ^i - ~2 [ -^r~ ] 2 ( -l) 2 It is clear that — f is an increasing function of o) for o) > 1. Thus, 4o) (o)-l) 2 it is of interest to find the value of o, > 1 such that — j = 1, i.e. 40) (o)-l) = 4o). The roots of this equation are 3 + 2/2. This implies that (I) 2 VU) , > 1 when o) > 3 + 2/2 (2.39) 4oi Hence, for a relatively large u) the magnitude of the eigenvalue n-i a, 2 d-o).) n (-1) II of n (S C ) grows rapidly as N-j becomes larger. i=j i=j+l Consequently, the error er grows rapidly according to (2.20). Example 2.2 Numerical testings of the 4-value Adams method with condition (a) or (b) ; (a) step size being alternated (b) step size being alternated once every three steps 29 Numerical testing "with conditions (a) and (b) are performed on the following two problems using the IBM 360/75. 1. y' = -y y(o) = 1 (2.U0) 2. y 1 = Ut 3 y(0) = (2.Ul) The initial step size and step size changing factor are taken to be 0.05 and 10 (or O.l), respectively. We are interested in comparing computed solution and exact solution throughout the time interval [0,5] • We summarize such results in Table 2.1, Table 2.2, Table 2.3 and Table 2.4. (A representative program (for Table 2.2) is given in Appendix A.) TIME COMPUTED SOLUTION EXACT SOLUTION 0. 72? 5 C. 5488 0.4168 0.3166 0.240b 0. 18 26 0. 1387 0. 10 53 0.8005 0.oT81 0.4618 C.335 0.362 C.390 .4 17 0.445 0.472 0.50C 0. 3508 C. 2664 C.20 24 0.1537 0.1167 0.8870 0. 6 73 7 2 73536D 1 163610 6201970 3 676940 084632D 8 3 52410 613L220 9 92 2 46 D 8312790- 0062630- 9 6 2333 D- *35410~0- 9097340- 191 145 0- 5191660- 8566970- 7139100- 94b 9990- 00 00 oc 66 Tie* 00 00" 01 01 _0_1_ n 01 Jl 01 01 02_ 2 Table 2.1. Integration of y' = -y, y(0) = 1 using U-Value Adams Interpolation Meithodwwith Alternating Step Sizes of .05 and .005 30 TIME 0.28000D 0.6CCCCD COMPUTED SOLUTION EXACT SOLUTION 00 0. 75578385720 00 00 C. 548811 «8770 00 0. 75578 5AB_ai 374150 CC 16361C QO O.87500D 00 0.41686234420 CC C. 4 1 6 8 e 2C 1 c 7D CC C.11050C CL 0..J31211186 5D DJ) 0. 33 121C ee22C CCu 0.142500 01 C.24C5C87576C CC 0. 24C5C846320 CO ._G.17000C 01 O*.l£26_8J?£0eJ.D_GC C. 1826 82524 ID _GC 0.193C0O 01 0. 1451484429C 00 . 1 4 5 148 1985C 00 0.225CQD ci Q JL ££52S2£2*t2n c.c c. i o 57qq?74 • 0. , n, r. COMPUTED SOLUTION 11 15500 7440-01 129 91923140 CC 575R7097^7P 00 2 1009461440 ~n\ 7f5 918f0130__0.1. 41636942660 03 138 7_8P2996_D. 05 4 7 3 C 993 5 530 C6 161PPA1726P C8 548 cn 3i34on nq 1867*^7 6 ?CO 11 6 3 5 9 5 P 3 8 3 3 1 ? 2\( 54^16^80 14 73736013720 IS 2 5107 57^4 10 17 85452876730 '1R 2 f ">1 l.C 8 62 2 00. ,.?_0 99124 31664D 21 _o. o. 0. 0. .p.. • ' . .0.. 0. 0. C. C. 0. _o_. 0. EXACT SOLUTION 1 1.1.5 6 64 0620-01 1296C0C0O0O 00 5861 81 64 06 ^0 17490C625C0 01 4,1234378919 .01 8 3 521 rroo n o 31 15714? 7S 390 02 2562890625O 02 40648594140 02 61 46 c; 6'\0C0O H2 81408844140 0? 12594^06 ?o 3 177676^7540 03 23134410000 03 3^787663790 03 3^213900670 03 4A a| 4 3 35716P__07_ 62 50C 000000 03" Table 2.3. Integration of y' = H , y(.0) = using U-Value Adams Interpolation Method with Alternating Step Sizes of .05 and .005 31 COMPUTED EXACT TIME SOLUTION SOLUTION 0.28000D 00 0.61506441670- •02 0. 614656CCCCD- ■C2 n.ftrrron ()() n. i ?9M7?3i9n nc Ot i pqisrnnnnnn no 0.8750C0 CO C.5662C21C1C0 oc c. 5e6l6164C60 00 £..11050C_CL. 0^149 C926.55.4CL _CL_ C1.45C9C2C51D XL 0. 1425C0 CI C.412347C543C 01 0. ,41234378910 01 _Q.O70aQD_Cl 0.P3 5214C8.7 9Q_.C1_ C. -£3521CC0G.Q0. OL 0. 193C0C 01 . 0.13874924930 02 0. 136748 8CC10 C2 n. ??«;nnp m n.?^f ?RqR B n,l + ' n-k+1 Vk' S n,k-1 ( = u, (2.U5) < k+1 > <* n " V/ § n,0 + (k+l) U n-1 " Vk } *n,l ^ + ...+ (k+1) (t n _ k+1 - t n _ k ) k 3 n>k _ 1 = Vl 33 where y. = 1 (t - t J 1 - (t - t v ) i n n-k n-1 n-k n-1 Let M. be the matrix A 2 (t - t , ) n n-k 2 (t n-l- W '• 2 (t . ._ - t , ) n-k+1 n-k <**><*» -W 11 (k+1) (t n-i - V k )k ••• |w)( WrV k » k ° and M. (i = 0, 1, . . . , k) be the matrix obtained by replacing i-th column of M A with the column vector A *k+l We claim that M is non-singular since A k-1 | M I = (-l)^ 2 (k+i)! n (t - t . ) A ._ ~ n-x n-k (t - t . ) ... (t , .. - t , ) n n-k n-k+1 n-k (t - t v ) k_1 ... (t _._ - t v ) k_1 n n-k n-k+1 n-k ^ Vandermonde Determinant 3>* Therefore, it is clear that 6 . can be uniquely solved by the formula n , l M. l n,i |M, i = 0, 1,..., k (2.U6) To make (2.U6) accessible for further analysis we need to express M in an alternate form, i.e. M. k 2 E h i=l n-i 2 I h . i=2 n " X k k (k+D( z h .r (k+i)( e h . r i-1 n_1 i=2 n - X 2 h n-k (k+l) h * , n-k Then _ , (i+2+...+k) M A i =1 ^:k i k h 2 E n-i 1 k h 2 E n-i i=l n-k h i=2 n-k i=l n-k h , i=2 n-k 1 1 2 k h k h . (k+l)( E lT ^r (k+l)( E -^r ... (k+l) Now let 6 i = h. h. - h. i l-l i-1 (2.1*7) Equivalent ly. 1 + 6. = i h h. l i-1 (2.1*8) (2.51) 35 Note that 6. may posses negative values. For i < k n-i _ n-i n-(i+l) n-(k-l ) _ /n . N , , . h h "5 "• ~h n (i + 1, q - k+1 P^l (1+ E n (1 + 6 n i^ P f ° r P > 1. ^k + l I-q J=i J (2.50) The determinant I M. I can be converted into a similar form. 1 i 1 Hence, it is clear from (2.k6) that g _ P i (4 n-l' 6 n-2"--' 6 n-k + l ) n ' i Wl> 6 n-2'"-> S n-k+l> where P. and Q. are polynomials in 6 -,,6 ^,...,6 , l l n-1 n-2 n-k+1 6 = 6 =...=6 = ¥ n n-1 n-2 n-k+1 the method reduces to the fixed step method. We have P.(0, 0,..., 0) 1 _ Q Q 1 (0, 0,..., 0) " i This means $ . is a continuous function of 6 ...... 6 , ,. in a n,i n-1 n-k+1 neighborhood of (0, 0,..., 0). Thus, for some constants d and D there exists a compact region U = {(6 n-1 Ul'I^Vl S n-k + ll D * } of the real (k-l)-tuple space on which 3 • is continuous. Moreover n,i I g . I is uniformly bounded in U . We shall state the above results in n,i' a lemma. When (2.52) 36 Lemma 2 . 3 The coefficients & . *s and 3 . 's in (2.1+3) are uniformly n ,i n ,i bounded provided for some constants d and D the step size change satisfies d ^„.i Wi D *" (2 - 53) Now we proceed to prove stability of the method. The next lemma is essential for that prupose. Lemma 2 . k Consider the linear difference equation k &_ = z „ i + h „ i ( z 3_ a z „ • + O (2.5*0 i=0 J m m-1 m-1 _._ n m,i m-i m' Let B, q, A, z be non-negative constants such that k E |3 . I < B ¥ m (2.55) 1=0 m ' 1 " h t 3 n <.i Then (2.55) is fulfilled if we put B = (k+l) B L. The remaining condition (2.58) in Lemma 2 .k is trivial, just take Uo v - max I Y ■ v 0 W" f(y( Vk'' W J) + h <-C , +? h k V k+2> (T >) n-1 n,k+2 n-1 n " Vl " h n01 (§ n,0 f y n-1 n,k+2 n-1 n where 5 . is an intermediate point between y and y(t ). Assume y is n,i n n bounded in [0,b]. Then it is obvious that Lemma 2.h is applicable here when we set 6 . = 3 . t J C . ) n,i n,i y n,i , _ a v,k+l (k+2), , A n " - C n,k + 2 h n -i y (x n } As analogous to the proof of Theorem 2.2, given some < q < 1, one can choose some suitable h . to satisfy condition (2.56). For the other n-1 conditions in Lemma 2.U, we need to set ~ ,k+l max I (k+2) , . , X * A = Ch maxt £ [0,b]l y 0(t) l where C is the uniform bound for C , ._ n,k+2 2. B = (k+l) B L q v - max | e I 0 0, e -> 0. max ' n ' Q.E.D. 1*2 Similarly, stability and convergence of the variable k-step Adams-Bashforth method can be carried out the same way. Thus, Theorem 2.2 and Theorem 2.3 also hold for Adams-Bashforth method. Example 2.3 Numerical testings of variable 3-step Adams method with step size being alternated. Again, machine computation is performed on the same initial value problem as in Example 2.2. The results are tabulated below in Table 2.5 and Table 2.6. (One program (for Table 2.5) is selected to be attached in Appendix B). TIME COMPUTED EXACT SOLUTION 69C734~3173~0 rc~ SOLUTION ~~Z~. 6 90 7 : 3"4T3C 60" 0.370000 00 0. or 0.6450CD 00 0. 52466252550 C3 00 0. 52466254210 0. 39851904H0 OC 0.92000D 00 0. 39851902360 or 0.1195C0 01 c. ,30270393720 00 0. 3C27039542D or 0. 1470C0 01 c. ,22992546950 0. 22992548520 oc 0. 174 50 01 0. ,1 7464 0.9475P543360 02 P. .13284925230 03 0. 18141126720 03 -Q__24 2 2 0.17 4 72 _0 3 C.?17l"?9ll06D 03 0,40 874 303 5 3D 03 0. 51769445R40 03 _0, 64 780 55 7660 03 Table 2.6. Integration of y' = kt , y(0) = using 3-step Adams Variable Step Method with Step Size Alternated with .05 and .005 From Example 2.2 and Example 2.3 it is seen that for Adams methods the variable step method is extremely stable. Even when subject to adverse condition (step size being alternated constantly) its convergence behavior is superior to the interpolation method with stabilizing condition (step size being held constant every k steps). It should be pointed out that the programs in Example 2.3 are written without explicitly taking condition (2.53) into account.* * Theorem 2.3 was proved by Piotrowski [12 ] , however his proof assumed without proof that the 3 . were uniformly bounded. n,i kk 3. STABILITY MD CONVERGENCE OF GENERAL MULTISTEP METHODS That the variable step method is more stable than the interpolation method in the case of Adams method is apparent in Chapter 2. We wish to generalize the investigation to the case of general multistep methods. Two independent proofs on convergence are given for the interpolation method and the variable step method, respectively. In section 3-3 numerical testings are carried out for the 3-step stiffly stable method in an effort to offer one more quantitative comparison of the interpolation method and the variable step method. 3.1 Interpolation Method In proving the convergence of Adams interpolation method, we make use of three essential conditions: (2.22), (2.23), and (2.2U). We observe that not all of these conditions remain valid when we are dealing with a general interpolation method. Fortunately, (2.2*0 still holds in the general case and that along with appropriate step size change constitutes sufficient condition for the convergence proof. Let h be the maximum step size, i.e. h = h . Let 9(t) be a max differentiable function in [0,b] such that ^0 < A <_6(t) <_ 1 for some A > (3.1) max |e'(t)| < 0' for some 0' > t [0,b] , h = h9(t ) V. n n Then from (2.1+7) >+5 h e(t. ) - h e(t. , ) 5 i TeTT-^ (3.2) e'(s.)(t. - Vl ) e(t i-l> S. £ (t. , t. ) 1 1-1 1 e ' (s i> h i-i = h e'(s. ) Thus, 6. is uniformly bounded by 6. | <_ h0' ¥ i (3.3) We shall express the step changing matrix C. in an alternate form. Lemma 3.1 The step changing matrix C. for a (k+l) -value general multi- value method satisfies C. = I + 6. C. l li (3.U) where C. = i o o Z (f) 5^ _1 1=1 * X Moreover, C. is uniformly bounded by some constant k~ > 0- 1 Vy Proof O U6 C = i o (w t ) O o {1+6 . )» = 1 + 6 o o L £-1 * J 1 + 6. C 1 1 By (3.5) 'I X k E k k ) is.r 1 < E (f) (h G'/" 1 *i ~ £=1 * hi Consequently for some constant k~ > 0, we have Cjl ±k 6 V-i (3.5) Q.E.D. The next lemma, follows from Lemma 3.1. Lemma 3 . 2 Let S and S be matrices which correspond to (2.l6) and (2.19) based on a (k+l) -value general multivalue method. Then S C _ = S + h . S m m-1 m-1 m where S is uniformly bounded by some constant k > 0. m Proof Analogous to (2.18) it is clear that S = S + h . S (3.6) m m-1 m where S is uniformly bounded by some constant k~ > 0. Substituting m S (3.M and (3.6) into S C . we have m m-1 S C _ = (S + h _ S )(I + 6 _ C _) (3.7) m m-1 m-1 m m-1 m-1 u " = S + h n S+6,SC +h 6 SC m-1 m m-1 m-1 m-1 °m-l m m-1 h e'(s ) = S + h . S + h , Szi_ s a m-1 m m-1 h , m-1 m-1 + h -i -| (3.12) Suppose we use a convergent one step method to start up the multivalue method. It is clear that as h +0, |e. T | I -> for all N such that max ' ' — N ' ' t N £ [0,h]. Q.E.D. 50 We like to point out that Theorem 2.1 is not a particular case of Theorem 3.1, as the two proofs are independent and distinct. In Theorem 2.1 it is essential to establish uniform bound for n (S C. ) whereas in Theorem 3.1 condition (3.1) plays an important role. 3.2 Variable Step Method Consider a k-step general variable step method £ (a . y , + h„ , 3_ , f „ , ) = i=0 n,i " n-i n-1 n,i n-i (3.13) of r-th order. We require that the polynomials 1, (t - t v )>. (t - t ) be exact in (3.13). For simplicity let us denote n— k s =t -t n9 m=0,l,..., k-1 n-m n-m n-k Then we have (3.1*0 1 . .. 1 1 S V4.T h 1 n-k+1 n-1 r r n , r-1 s ... s , , ., rh n s n n-k+1 n-1 n h h n-1 n-1 r-1 rh s n-1 n-k+1 — — -k /s a n,0 . . • n,k § n,0 . . n,k _ l = (3.15) According to the algorithm for variable step method stated in Chapter 1 (page 11), we convert (3.15) into the following (r+l) x (r+l) system. n-p 1 n-p 2 ... 2 2 s s n-p 1 n-p 2 n-p 1 n-p 2 ^ where n-1 . . 2h , S n-1 2h „s n-1 n-q " n-1 n-q r-1 r-1 rh , s rh , s n-1 n-q n-1 n-q M. 51 n»Pn n >Po n,q. n,q r+1-2, n,0 n,l n,r (3.16) ° - P l < P 2 < * ' * < P £ - k 1 q i < q 2 <...< q^ <. k and b .'s are polynomials of h ,»•••» h , and the specified a .'s and n,i * n-1 n-k * n,j n,j 's for i = 0, 1,..., r. From (3.1*0 k 5 E h . n-m . , ., n-i i=m+l (3.17) Then it is readily seen that 1 A ' n-k Q A (6 n-l' 6 n-2'---' 6 n-k+l } where Q is a polynomial in 6 -, » ^ 5'.'5 <$ , ,-,• The existence of unique A n— 1 n— <— n— K+J. solutions for (3-l6) in the case of fixed step method implies that Q is nonvanishing in a neighborhood of (0, 0,..., 0). Hence, for some constants d and D there exists a compact region "■ {(6 n-i---' W'^Vi 6 n- k+ i ^ * } (3 - l8) on which M is non-singular. This assures the existence of unique solutions A for a , . . ., a ,3 , . . . , 3 n,p 1 ' n,p £ n,q 1 ' n, Vi-l-Ji in (3.16). 52 We shall show that these coefficients are rational functions of 6 n-l'"-> 6 n-k+l' Lemma 3.3 Let 6 .,..., 6 ... satisfy (3.18) ¥n. Then the coefficients n-1 n-k+1 G „,...,& , » 3 «»■... p , can he represented in the form n,0 n,k n,0 n,k (0, 0,..., 0) i 6 . =...= 6 • . = a i n-1 n-k+1 07 (o, o,...,o) ^ tt i (3 ' 22) 53 Apply partial fraction technique to get P - a. Q a . i a . = a. + n,i i Q a i In view of (3.22), P -a. Q is a polynomial which vanishes at (0, 0,...,0) i i Hence P - a. Q does not contain any zero-th degree element. Thus, a. l a. l l (3.19) is verified when we set P - a. Q a. 1 a. R = — ^ (3.23) a. 0, la . I < K_ ¥n. Take a. ' n,i' — a. 1 1 ■mo ■y I I K- = n . ' n {K^ }. Then a .is uniformly bounded by K„, a l > 0. n,i B Let K 3= K a +K B L (3 - 3l) A = (k+1) Kg L (3.32) Then Q . and A are uniformly bounded by K n and A, respectively. The n,i n 3 above findings are summarized in Lemma 3.^- Let the step size change be such that (6 ...... 6 , _ ) e U n-1 n-k+1 for all n. Then a k-step general variable step method (3.13) can be converted into (3.28) with 3 • and A being uniformly bounded. n ,i n We state two useful lemmas from Henrici [9]. The first one is the same as Henrici 's. Lemma 3 . 5 k k— 1 Let the polynomial p(c) = a n ? + a i E +...+ a satisfy the condition of stability (namely the modulus of no root of p(c) exceeds 1, and the roots of modulus 1 be simple) and let the coefficients y (£=0,1,2,...) be defined by 55 1 = Y + Y n 5 + Y C 2 +.. . (3.33) a„ + a, E +.. .+ a C^ •o - 1 ^ + -.- + \^ Then r= SUP |v I < « £ =q ,1 . . . ' h ' The second one is subject to slight modification. We will offer the justification for such change in Lemma 3.7 • Lemma 3 . 6 Consider the nonhomogenous linear difference equation a_ z + a n z _+...+ a. z m 1 m-1 k m-k = h {3 z +...+ g z , } + h , A (3.3*0 m-1 m,0 m H m,k m-k m-1 m Suppose the polynomial p(?) = a ? +...+ a satisfies the condition of stability, let B* , 3 , and A be non-negative constants such that (for m = 0, 1, .. . , N) |B J + |8 J +...+ |3 J < B* (3.35) m,0' m,l m,k' — m,0' - (3.36) A J ■ A (3.37) nr 0 (3-38) — m-1 — Then every solution of (3.3*0 for which satisfies zj 1 Z u = 0, 1,..., k-1 (3.39) z | : K* e L * (t m " t ) (3.U0) m where 56 kArz + TA(t - O K* = — i£ 0- (3.H1) 1-q L* = g- (3.U2) A = |« I + \ a ± \ + --- + l\l (3.U3) Lemma 3 . 7 Let z satisfy m m-1 Then z I < K* + L* Z h. z. (3. MO m 1 — . _ l ' i 1 i=0 z I : K* e L * (t m " t ) (3.U5) m 1 — Proof We shall show by mathematical induction 1. m = 1; from (3 .kk) \z ± \ <_K* + L* h Q |z Q | <_ K* + L* h K* by ( 3 . Ul ) = K* (1 + L* h ) L*h < K* e = K* e 57 2. Assume (3.^5) for m<_n. From (3.UU) n n L* (t,. - t n ). = K* + L* L h. K* e X U i=0 x n L* (t. - t) = K* (1 + E L* h. e X U ) i=0 x n L* (t, - t.) n L* (t. - t ) = K* (1 + S (1 + L* h. ) e X U - I e X ° ) i-0 X i=0 n L*(t. +1 - t ) n L* (t. - t) ji K* (1 + £ e - E e ) i=0 i=0 = K* e Thus, the induction is completed and the claim made in (3.1+5) is verified. Q.E.D. We proceed to show the stability of (3.13) Theorem 3-2 A k-step general variable step method (3.13) is stable provided the step size changes satisfy condition (3.l8). 58 Proof Let B* = (k+l) K 3 (3.1+6) 3 = K 6 (3.U7) Y = max ly I (3 kB) 0] = y(t n ) - a x yft^) ...-o, k y(t n _ k ) - Vl (S n,0 «**.» + --- + S n> * f ^ (t n-k !)) + Vl \, = 0(h r+1 ) (3.50) max since (3.28) is of r-th order. Define e = y - y(t ). Combine (3.28) n n n and (.3.50) in a similar manner as in Theorem 2.3, we have e = a., e , n 1 n-1 + ...+ a. e . + h . (| _ e +...+ g e , ) + h _ I (3.51) k n-k n-1 n,0 n n,k n-k n-1 n where 6 .=3 . f (c • ), C .an intermediate point between y and y(t ) n,i n,i y n,i n,i * 'n n I h n-l . * Wl„C«)f. ) ,*V-1 i ( v Ste!)**,, yt^'fx ) h , k k h /„.. \ _ _n=l v ( i n ~'T) r B Y l ; (T ), x . , x . e (t , , £ ) r! .\\\h / n,i-l y n,i-r' n,i' n,i n-k' n' x=l j=l n-1 We set i.= U +1 )K 6 L, 6=K 6 L, E = <^-li e il ((3-52). (3.53), (3.5M) r max i r +1 1 « _ /■ o c q \ A=Ch , pf ,,i y/+J» for some C > 13.55; max te L0,d J it ; Choose r according to Lemma 3.5- Given some < q < 1 choose h^ according to (3.38). By Lemma 3.6 we have, at time t^, TB* 1 N ' — 1-q where A = 1 + | ou | +...+ | ot | . Again, assuming the use of convergent single step methods to obtain initial points. Then it is clear that as h -»■ 0, lej ■* 0. max ' N ' Q.E.D. 6o It is clear from the preceding theorems and lemmas that the stability and convergence of the variable step method depends on that of the corresponding fixed step method. To begin vith one must choose a stable and convergent fixed step method. Conditions (3.1) and (3.l8) will both be fulfilled if the ratio of successive step size satisfies (1.25), namely, h r- h _ n n "- = 0(h ) h ., max n-1 Intuitively, this means that interpolation method and variable step method are both stable under mild step size change. The generalization of Theorem 3.3 to systems of differential equations can be readily accomplished by replacing y with a vector y , absolute norm |* | with vector norm | |* | | . The final form for error bound will accordingly be the same as (3.56) with the corresponding constants A, T, . . . . 3.3 Numerical Testing of Stiff Interpolation Method and Stiff Variable Step Method In this section we shall perform numerical tests on a multi- step method other than Adams method. We choose a' stiffly stable method because it is a very useful numerical method in dealing with stiff differential equations. Furthermore, the associated linear difference equation of stiffly stable method is quite different from that of Adams- Moulton method although they are both implicit methods. k y = I a. y . + h R f (3.57) n . ., 1 n-i n i=l is fixed k-step (k<_6) stiffly stable method 6i (Definition of stiff stability can be found in Gear [8]). The corresponding interpolation methods and variable step methods for k = 2, 3 are contained in the following examples : Example 3 . 1 The representations of interpolation method and variable step method based on the two-step Ada.ms-Bashforth predictor and the tvo-step stiffly stable corrector . (a) Interpolation Method The corresponding matrices. A and i_ of representation (l.lO)-(l.ll) are 111 12 0.1 I = 2/3 1 1/3 (b) Variable Step Method A-B Predictor *n,(0) " Vl + Vl <»£i Vl + ^2 W where f . h + 2 h (p) _ n-1 n-2 n,l 2 h n-2 g(p) _ Vi p n,2 2 h n-2 Stiff Corrector y n,(m+l) *S y n-l + «n,l *n-2 + Vl § n!o '„.<*) 62 where a a (c) n,l (C) n,2 ( (C) n,0 n , ( m) Vl ^ \- 2 ' h n-2 S Vl * h n-2> h n-2 (2 Vl + h n-2> ( Vl + V2> 12 Vl + \-2> Example 3.2 Numerical testing of Example 3.1 Machine computations for the methods in Example 3.1 with alternating step sizes are carried out on the initial value problems (2.U0) and (2.Ul). The results are summarized below in Table 3.1, Table 3.2, Table 3-3, and Table 3 .k. TIME COMPUTED SOLUTION EXACT SOLUTION 0.325 C.6CC CCD CC CCC CC 0. c. 722 546 4313 6622 332C CC ?<6C CC C. C. 7225273 5488116 536C 00 361C CC 0.875 0.115 ccc ccc CO CI c. c. 416 316 69C4 4623 718C CC 63<=D CC C. 0. <16£62C 3166367 1S7C 694C CC CC 0.142 C.17C 5CC coc CI CI c. c. 0.197 0.225 5CC CCC CI CI C. 0. 24C ii£ 136 1C5 3424 53_2C 626c 2824 S530 216C CC CC C. c. 24C5C64 1826635 16CC OC 667D CC 0. C. 13 6 7613 1C53992 632D CC o 01 ci 0.41 35501 59RQ 0. q* 6^2 794- SO 01 1 0.4 123437* 9 in o. e^nccccoo 01 01 0. 197 son 0.2^009 01 01 0.152 vu«453n n . / q a. r. n ? 1 1 s ^ 2 0. 15214P753 90 -] m 2 5 6 2 a Q (■•> 2 5 02 r '2 0, 2t?600 0.7 3 0000 CI 01 0. 3 0'7 q ^9 . "3 * K ) lV O.^^O") CI 0. 406*6 8 3 65 20 0? 0. . h. 1.5.1 ?_6 64 4 jQ Q2 7 9 ■- . ^z,c *,* 4 7' 0. 17'v): I 97660 12 0_3_ 0.17? 7" 5 07 770 3 - ? ^ i 4 i n as > ?o r^ 0. 4QA4 f n594l40 C2 0. 614 S5 600 Q2 "> . ; ' 9 4 " M a 4 a i a. o ~ 2 ._! 2 69 4 4 5 06 20 Q 0.1726 7602540 03 0.417500 01 3.445 Q.,n Q l 0.4 7 7 5~0 fl 0.5QijO;)Q 01 . "* 9 n 3 1 r- 5 f- a 3 0. ^q?2 c '° ^Q9Q0 Q 3^ ~.4s*t56H~«5in 9-3 0.625 1^0771 l n 0* 0.3 02 9.2 6 6*790 9 Q. J 9~>1 9 900 6 2 03 n. 49R43 351169 2 0. 6250C0000Q0 03 Table 3.3. Integration of y' = Ut , y(0) = using 3-value Stiff Interpolation Method with Alternating Step Sizes of .05 and .005 6U TIME . 3? ^r ^n no o.6 00ooo~~ooT 0. H7^n v )n 00 01 COMPUTED SOLUTION ).Po^7 ,r i^in oo _n.j_H'*^7i^^^ oo 0. L 7 «^ 61^7^0 0! 0.4 I.' 3S?1 15 70 0] EXACT SOLUTION _0, 1 1 lS6ft4C6 ?0-0 1 0. 1 2 9600000 O^ 00 0. sqm P] 44050 np 0. 17^''0C^? c -0^ "ci 0, <* 1?3^^7H^10 01 0.17 0'' 10 01 0.197*00 01 o.??5co*n 01 Oj25?^lD 01 o. ^?f.n7^Pi^n oi 0. l r ^?6S345?0 0? 0, H 1^21 VK *n ."1 0. 15?14fl75?9n 0? O.?«0000 01 0.3 07* on 01 0.25*441 175 0.4C667763? 0.61 4*9 1 $44 0. fH}4"*7301 4 ^n o? o.? e ?6?»")0625n o? RO 02 0. A064R5941 '.0 0? AO 0? O.M44550000O 0? in Q2 0. 39403344140 02 . 3 3 5 ~ ^ n n . 1 2 ^ 7 P > " ) ° ^ Qn 03 0. 12 5Q445C620 3 0. 3 6? K on 01 . 1 7 ? 7 1 5 6 n n 03 0. 17267602*40 0* o.** Q oooo 01 0. ?3 1 ">P':o ?o c n °3 r,21l i44100n(i 03 0.4175 o n 01 0.303R7'321 i an 0^ 0. 303.a26A.R79n 0^ 0. 445C0n 01 0. * c 2l9R6O06 n 03 0. 3 3 2139Q0'-?n 0^ . 4 7 ^ 5 r n r\ 0.4QPS^OP^4rn 03 o. 49«43^531 60 r 3 0.50000n 01 0. 626075^7 7°n 03 Table 3.k. Integration of y ' = kt , y(0) = 0. 6?50C00C03O 03 using 2-step Stiff Variable Step Method with Alternating Step Sizes of .05 and .005 Example 3.3 The r epresentations of interpolation method and variable step method based on the three-step Adams-Bashforth predictor and the three- step stiffly stable corrector . (a) Interpolation Method The corresponding matrices A and l_ of representation (l.lO)-(l.ll) are A = 1111 12 3 13 1 L = 6/11 1 6/ll l/ii 65 (b) Variable Step Method A-B Predictor Already presented in Example 1.2 Stiff Corrector For simplicity we shall determine the coefficients in terms of time intervals. y / +1 x = a (C i y i + « (C o y o^ Cc l y » + h i & {c l f t \ *'n,(m+l) n,l n-1 n,2 n-2 n,3 n-3 n-1 n,0 n,(m) where a (C n } = (l/|M.|)(t -t )(t -t J 2 (t -t )[(t -t J(2t -t _-t _) n,l 'A' n n-1 n n-3 n-2 n-3 n-2 n-3 n n-2 n-3 " ( Vl-V3 )( V2-V3 )( VV3 )[( V2-V3 )(t n-V3 )( VV2 ) " (t n-l- t n-3 )(t n- t n-3 )( V t n-l ) + (t n-l- t n-3 )(t n- 2 - t n-3 ) ^n^n-l 5 ] |M. | = (t -t . )(t _-t ,)(t -t _)[(t -t _)(t -t ,)(3t -2t -t ,) 'A 1 n n-1 n-1 n-3 n-2 n-3 n-2 n-3 n n-3 n n-2 n-3 - (t -t _)(t -t _)(3t -2t -t J + (t _-t _)(t -t _) n-1 n-3 n n-3 n n-1 n-3 n-1 n-3 n-2 n-3 ( -Vl +t n-2>] f = f(y . ., t ) n,(m) 'n,(m) n 66 Example 3.^ Numerical testing of Example 3.3 Again, machine computations for the methods in Example 3.3 with step size "being constantly alternated are carried out on (2.U0) and (2.Ul). The results are summarized in Table 3.5, Table 3.6, Table 3.7, and Table 3.8, COMPUTED EXACT SOLUTION SOLUTION TIME 0.325COO 00 0.72252631980 0. 72252735 ISC 00 0.600C00 0^ C. 54F8556493H 00 C. 54 88 1 16 36 10 00 (K875C00 00 .415 462143 40 00 0. 4 1 68 62 C 1 9 70 CC 0.115C00 01 6. 3616<52 < ?0"C3D 00 0.31663676940 CO" 0. 1 4 2 5 0_ 01 - C . 12C9 0_5 C4840 C I 0, 2 4C5C846370 CC 0.17 CO 00 01 0. 46 813 9 29 160 2 0. 1826 83 52410 CC 0.1975CO CI -0. 150C ?7 C ?56D C4 0. 1 3876 1 3 122C CO 0.225000 01 0.48272478480 05 C. 1C 53 9 9 22460 CO 0.252500 01 -0. 15530525940 07 . P005 8 3 1 2 790-C 1_ 0~.28CC0O"01 C.49~96590096"6 C8~ 0. 6C810062630-6i 0.107500 01 _-0.16CJ_5^ Q C4CO_ 1C 0.46 18962 F3 PO-C 1 "6. 335C0C ! 01~ 0V5 17 188 42 330 11 0.35084354100-01" _?_»_L ft 25 r O 1 -0. 16639*49070 13 0. 2 6 64 qp C7340-0 1 0.390000 01 0.53533781770 14 0.20241911450-01 0.4175C0 CI -0.17223103 220 16 0_._15 375 1 9 166C-01 _ 0.445C00 01 0.5.541.1376750 17 0. 1 1 678 566970-C1 _0._4725C0 CI -0, 17_8?73?7_120_19 0., F F 707 1 39 10D-C2 C . 5 C 10 C 00 1 '" C.' 57 35 5 3 6 C 47 o" 2 C 3 . 67 3 7 946 99 90 -C 2 Table 3.5. Integration of y' = -y, y(0) = 1 using U-value Stiff Interpolation Method with Alternating Step Sizes of .05 and .005 TIME c.37cccr cc 61 COMPUTED SOLUTION c.e < ?c7? c ;667er cc EXACT SOLUTION 0.6<5C73*??C6C CC 0.6*5CCD CC C. c 2*66*?PC c O OC C. e i2*662S*2 o . 9 2oncr co c^ c p57C c ic c p cc c. 2<;f51<;c*i 0.1 10 0. 1*7 sec CI cc^ci 1C in CO cc c. r . ?C27C57Slin OC 7 ? c c ? "i 1 c 7 2 C r C C. r . 2C27C 22<^2 355* 5*85 2C 20 CC CC 0.17*50C 01 C. 17*6*652 < 5 c ;0 CC C. 0.2C2Cpn CI .C. l?7fe56PlP3r OC c. 1 "5464 1 3 2 6 5 CD CO IC cc 0.229 0.257 c cc r i CCC CI 0. \CClt2 cc . cc X CC 76 536537CCn-Cl C. 1CC76 76 c 35 I 3 c 3 5*?* 3C CC 20-C1 C.2P* 0.31? 5CC CI rrr, f\ C. r . 5613^5 555 **1 C 7P62 16 r-ci n-ri 0. 5 p 1 2 * *. * 1 5 7 2667 16P* *C-C1 2C-C1 0,3?9 c or CI C.335*l!?732r-01 0. C.' i 67CCC CI 0.2**76^^3^-01 Q. 3 3 5 AC 2 5*76 55*1 *6<=5 70-C1 5C-C1 0.394 500 CI c .*?2ccr c 1 C. lc3 c 16CC36n-Cl C. o.i*ftq» ct ^6*2r-oi c. is^5i i*6<;e 2163 6** c 7C-C1 CC-C1 C.**9 Q.*77 5CC COJL CI C. 111649^2920-Cl 01 C.fi*fC5e' a *P?n-C2 C. 1116* F*K2 6FC6 6 C 1 6 2 C - C 1 CC-C2 C.5C*5CD CI 0.6**1623675f-C2 0.6**1*6C36*C-C2 Table 3.6. Integration of y 1 = -y, y(0) = 1 using 3-step Stiff Variable Step Method with Alternating Step Sizes of .05 and .005 TIME C.32 C.6.C 5C0O cc cr cc rr 0. 11 ".11 COMPUTED EXACT SOLUTION SOLUTION 134520920-01 0.111566*C62C-C1 ^ccc7A?;n "c 0. 1296.rcH00 r > p CC 0.R75CCC CC 0.5597P52216C CO C. 5 5 6 1 f 16*C6C CC 0.115CCC CI C. 2*^*22*1 CCC CI . 1 7*^0 C62 5CC CI 0.1425 CO r l -C.!6F=M'75 c n C2 C*123*°7f ( nn CI C.17CCCC CI C.60C70' a 6-g31C C * O.P3 c 2 1CCCCCC CI C.19 0.2? 7 C CD c CQr CI -0.1666571 5160 r 5 C. ci c .*7CP g ;6* c 7^n C6 c. 15 21*F 62£5 753S C 62 5 C C? n C2 0.25 0.?* 25C0 rr oo CI CI •0. 13 f.37 27 c 8?^6*^ CP C. *0**P c S* 1*C C2 *3~' a 7 6''2n rc : C.614656CCC0C C2 0.^0 C.33 7^or 5 CCO fi.3625^0 c.3qccc r CI r i X_L - • 10557226 P 3 C C. 2 C 75*C 73*PC . r # p "3 p p c /j C A T «= 11 12 C. 0. PS 12 I 3 15 0. 17 *CPP 55** 2 6 7 6 13** ** 1* 5C62 p 25* 1CCC C2 C_C3_ r £3 C C? C.*l 0.** 0.*7 C.5C 7^cr 5mn CI C L 0.6 6^ C 5267CC0 1ft 0. 0.1PPCl*' a 7?2n IP c . ■a r — ^ - c 2526 7135 6P7C CC62 r c? C C 3 7 C CC CCCO CI CI CI* CC9'»1635r IS 0.*SF*225?16 § * 5600 KH 21 C.62^CCC r 0C" c? Table 3.7. Integration of y' = ^t 3 , y(0) = using U-value Stiff Interpolation Method with Alternating Step Sizes of .05 and .005 TIME C.37CC CP 5 CC C cccc rr C.6A C.<3? CC CC 0.11^00 0. 1A7C0P CI 01 68 COMPUTED SOLUTION 0.167F 7UC0 7CC0 Cl CI 0.40 C, 51 F 2 '♦ "* 5 76C«=^ AR9C 03 C3 0. r 4CF24 517*c 3C35 445* 3C AC C3 r ? 0.50ASOO Cl 0.647FC617540 C3 C.647fiC5576fn C2 Table 3.8. Integration of y' = Ht , y(0) = using 3-step Stiff Variable Step Method with Alternating Step Sizes of .05 and .005 Programs for Table 3.5 and Table 3.6 are attached in Appendix C and Appendix D, respectively. Based on the above numerical tests, the stiff variable step method "seems to be" more stable than the stiff interpolation method. However, one should be cautioned that this is by no means a quantitative conclusion. This writer feels that further theoretical research along this direction may eventually lead to a clear- cut answer, one way or the other, such as in the case of Adams method, to the relative merits of stiff interpolation methods versus stiff variable step methods . 69 k. PRACTICAL CONSIDERATIONS OF THE STABILITY AND CONVERGENCE THEOREMS Step size varying is not merely of theoretical interest as further understanding of the mult i step methods is enhanced hy a knowledge of the stability criteria. Even more significant, it is of practical use in that computer time is saved hy reducing the number of steps "while still achieving a desirable accuracy. Consider the error bound (3.12). We write I KM i4 x) + Ki 2 Z) e {h - 1] where 4 I} = K S M e H {h - 2) t t \ -i K K ( t. T — t - ) K (D = I (l+e B I 0, {k3) *- K Although the constants K and K are clearly independent of the step sizes taken, they are not computable for most problems. Nevertheless, e is closely associated with local truncation error. In practice, we shall make use of this latter property to control the global error cl . Assuming the exact solution is sufficiently dif ferentiable, there exists a smooth function _(t) called the principle error function such that the local truncation error satisfies ^ - ± (U - U > for a r-th order method. Given some specified e we choose the largest possible h . such that n-1 |d | z h e 1 — n ' ' — n-1 according to (2.30). We shall show that such a choice for h , does not n-1 violate condition (1.25). 70 From ( U . U ) we have lli(t ntl )|| h n r+1 + o(^ +2 ) 1£ h n (U.6) If we choose the largest h and h so that these are equalities, h -h . . ||l(t + .)|| h r+1 + 0(h r+2 ) - H^t )|| h^ 1 + 0(h r+2 ) n n-1 . 1 ' ' *- n+1 ' ' n n MI - n ' ' n-1 n-1 h " "" e h . n-1 n-1 • -(I l±{t *i)|| » ^ + 0(« hr+1 ) - I liK* )|l ^ i + 0(h r+ ^)) E "- 1 - n+1 M nn nn ' ' *- n ' ' n-1 n-1 : 0(h ) — max Therefore, when the local error is controlled, the step size ratios satisfy the condition for stability and Theorem 3.1 shows that the global error is bounded and converges as e goes to zero according to (U.l) and (k.2) The same thing can be said for the variable method since the error bound (3.56) can be converted into the same form as (U.l). The step size changing mechanism is also similar, which we will not repeat here 71 LIST OF REFERENCES Alfors, L. V. COMPLEX ANALYSIS, 2nd ed. McGraw-Hill Book Co., New York, 1966 . Bartle, R. G. THE ELEMENTS OF REAL ANALYSIS, Wiley, New York, 196U. 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 , 60_, #1, pp. 98-108, 1972. Cheney, E. W. INTRODUCTION TO APPROXIMATION THEORY, McGraw-Hill Book Co., New York, 1966. Collatz, L. THE NUMERICAL TREATMENT OF DIFFERENTIAL EQUATIONS, 3rd ed. Springer, Berlin, i960. Coddington, E. A and Levinson, N. THEORY OF ORDINARY DIFFERENTIAL EQUATIONS, McGraw-Hill Book Co., New York, 1956. Dahlquist, G. "Convergence and Stability in the Numerical Integration of Ordinary Differential Equations," Math. Scand. , ib PP- 33-53, 1956. Gear, C. W. NUMERICAL INITIAL VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS, Prentice-Hall, Inc., New Jersey, 1971. Henrici, P. DISCRETE VARIABLE METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS, Wiley, New York, 1962. Hoffman, K. and Kunze, R. LINEAR ALGEBRA, Prentice Hall, Inc., New Jersey, 1961. Nordsieck, A. "On the Numerical Integration of Ordinary Differential Equations," Math. Comp. , l6, pp. 22-1*9, 1962 . Piotrowski , P. "Stability, Consistency and Convergence of Variable K-step Methods," Conference on the Numerical Solution of Differential Equations , 109 , Springer, Berlin, 1969- Van Wyk, R. "Variable Mesh Multistep Methods for Ordinary Differential Equations," Journ . Comp. Physics , 5_, pp. 2UU-26U, 1970. Varga, R. S. MATRIX ITERATIVE ANALYSIS, Prentice-Hall, Inc., New Jersey, 1962. 72 APPENDIX A PROGRAM FOR THE INTEGRATION OF y' = -y , y(O) = 1 USING U-VALUE ADAMS INTERPOLATION METHOD WITH STEP SIZE BEING ALTERNATED ONCE EVERY THREE STEPS 73 A-VALUE ADAMS INTERPOLATION METHCn FCR CY/CT=-Y C STEP SIZE INVERTEC EVERY THREE STEPS IMPLICIT REAL*B( A-H,C-Z ) DIMENSICN AP ,ACC(A) ,6P<-T) PRINT 920,T f YN, YEXACT Q?n FrPP AT 2 , . Y N 3 " R E f HE VALUES OF Y N,N-1 , N-2,~~N-3 OVER THE TIME YN=YN1*mni ■*< hpi -F ( YN1, T-HNl )+RP2*F(YN2, T-HN1-HN2) + «P3,cf (YN3, T-HNl -HN2-HN3 ) > C LET PCS, PC1,RC?,RC3 RF THE COEF F I EC I FNTS CF- A-M CORRECTOR A=HM2 /HNll R = RC ( HN2+HN3 ) 3=( l.PO+2 /HN1 .00-- A )/(12.00~B-(i.D0+B)* E N S I r N 'A P ( 4") , S C (4 ) , S C C ( 4 ) , R (A j ," C < 4) F(Y,T)=-V __ ~ H=C.0 5CO C N TS THE TOTAL NUMBER OF STEPS N = 2 T=2.0C*H ALPHA IS THE STEP CHANGING FACTOR AlPHA=C.in^ ESfAPLTSF t>F NdRSrECK VECTOR AT" THE INITIAL PCTFT SCC( 1 )=PcXF(-0.1CQ ) scc( ?)-h^(-dexp( -c.ino ) scc(i)=o.5^F^H^nExp(-c.ino) SCC(4)=( 1/6. OC |*H*H*H* (-CEXP(-C.IDO) ) _PRINT °00 ~9"00 FC"R v ATT/77 X ,"1 FT 15 X , 2>YN \WX , 7"FEXTCT'~Y 15X> C R IS THE NHRPSIECK VECTfR AFTER STFP CHANGE 100 P(1)=SCC(1) B(2)=ALPHA*SCC(2) B(?)=ALPHA*ALPHA«-SCC(3) P(4)sALPI-A*ALPHA*ALPHA+SCC(4) AP IS THE NORCIECK VECTOR AFTER PPECICTION ap( i) =r( i )*p( 2i*ei ^i*ehi AP(2)= P(?)*2.CO*P(3W3.ro«P(4) AP(3)=B ( 3 )+3.rC-P( 4) AP(4)=B(4T H=H*ALPHA T=T + H C(I) IS TFE AMOUNT TO RE CCRRECTEC IN THE CCRRECTCR G=H*F( AP( 1 ) ,T)-AP<2 ) C( l ) = (6. nc/n _ c r-)*c C(2)=G C(3)=(*. 00/11. CO )*G C(4)=( 1 .DO/11. DO)*G SC IS TFF NCRCSIECK VECTOR AFTER FIRST CORRECTION HO 110 1=1,4 11 Q_ SC(I ) = AP( I ) +C ( I ) ___ GC = H*F ( SCTT) , T f-SC ( 2 ) _CJ1) = (6.DC/11.C0)*CG C( 2 ) = GG "" C(3)*U .cc/u .no >*co C(4) = ( i .nc/n.oo ) *cg C SCC I_SJFE NHR.DSIECK VECTOR AF TER S ECONC CCRPECTICN DC 12 J = l ,4 12C SCC(_-SC( J UC( J) r change" t hf "step size ALPHA=1 .OO/ALFHA C PFCOPO THE INTEGPATICN HISTORY EVFRY 10 STEPS IF( (N/IO.DO-N/IO.OT.O.CICO) CCJTO 15C ~YN = SCC(iT ">" JT E X A CT= C E X P < - T )_ _____ PRINT 920tt f YNt'YEX ACT* 80 920 FDRWAT(C1^. 5, 2020.10) 150 IFp(-r ,r c n r ) y?-rfxp(-o . lcno ) r n fl Ti-P tct/u nlFREP cf stfps C T I <= T H^ CLRPFNTTINF AI.Pt-A IS N=2 7FF STFP rt-ANGING FAC7CR t = ?.dc<-f ALP»-A = o.ino 7 LPT t-Nl ,i-\2 , h N3 PE 7FPFF CCNSFCU7IVE STEPS CVER THE TIME f INTFRVAL TN1»TN2,TN3 ^_ LFT FFl,FP2tFP* HM=O.C=CC FE TFE CCEFFICIENT CF A-P FPECICTCR l-fOrCC^C HN^=0.05CO C LET YN ,VM ,>N?,VM PF THE VALUE CF Y CVEP THE TI*E C INTERVAL TN » TM f TN2 » TIS 3 Y M = Y 2 YN" = Y 1 YIS3-YC fC w Fl T F TF-E Pc-FT ICT FT VALLF CF Y mc RPl = l.n: + (HNlM?.C r< FM+6.C r, *FN2 + 3.n*FN3))/(6.rr-'-FN?*(FN2 + HN' pp7^-(hM^(2.CC : FM^ Q .CC^FN2^ a .nC^FN3)W(fc.CC^FN2^N3) pp^ = ((-M^(2.C0 J; l-M43.C0^FN2))/(6.nC*FN3*(FN2* F N 3 )) T=T+FN1 TN = T TM = TN-HM 7N9 = TN 1-HN? tn^=t N2-fis q YN = YM+HM"(RFl-F(YM,TM»+nF?*F(YN2.TN2l + PP3*F(YN3,TN?)) LFT SC0tSCltSC2»SC* 3F 7F": CCEFFICIENTS C F THE SIFF CCPPECTCR nFL = ( TN-TM)^(TM-T(\ :, )- > (TN2-TN?)«-((TNZ-TN3)^{TK-TNl)^ C (^tn-2-7N?-7N3)^TM-TN'»)»(T^TI^3>'M3*TN-2-' > TN1-TM-M + ~C ( TM-TN? r*l TN2-TN3)'»-TM + TK2) ) S C C = ( 1 / C F L IMTM -TM)»CTN2-TIS3)MTN-TN31»((TN?-TN?)* c ) •(?'■• TN-7N2-TN?)-(TN-TN?)»»?) SC? = (l/r. cl)-'(T\-TM)*(TN-TN?)**2*(TM-7N?)* f (-(7M-TM)* (?*TN-7M-7N3)-H7N-7N3)*»?) sc. a =i . nc- c c i-sr ? C r F P P C 7 7 F c VALUE CF Y TUICF YN = sci'-YM + $c? y 'VN? + $c3*YN3 + FM*scc »HM*SCC*F(YNt7M PECCRF 7FF TN7EGPA7ICN FlSTfRY EVERY 1C S7EFS ALCNC 7HF WAY ip( (N/ir.nr-N/in > .f-T.c.rirc) gc 7c 15c YFXAC7=CFXP(-7) P P T N 7 c?c,7,YN,YFXAr.7 83 c?C FCP»'AT(ri2.'5,2C?C.lQ) 1 5 f IF(T,G7.«,rCI nc TT ?cc C 1MTI AU7E ThF HATA NECESSARY FCR THE NEXT STPP hm=hn? hN?=HM FM=Al PhA*HM Y>^ = YN2 V N *» = V M YM=YN N = N + 1 CHANGE ThF STEP' SI7F ALPM = 1 .PC/ALPHA GC TC KH 20C CC^TINLF STfP FNT 8U VITA Kai-Wen Tu was born in Mui Yuen, Kwangtung, China, on May lU, 19^6. He obtained a B.S. in Chemical Engineering from the University of Wisconsin, Madison, Wisconsin, in 1966. After that Mr. Tu spent a year and a helf in industry as a research engineer. In the spring of 1968 he entered the University of Illinois where he received his M.S. in Mathematics in June 1969. He held appointments as teaching assistant and research assistant, mostly in the former capacity, in the Department of Mathematics, University of Illinois, from 1968 to 1972. S OGRAPHIC DATA 1. Report No. UIUCDCS-R-72-526 3. Recipient's Accession No. 5. Report Date July 1972 r e and Subtitle STABILITY AND CONVERGENCE OF GENERAL MULTISTEP AND MULTIVALUE METHODS WITH VARIABLE STEP SIZE 6. K nor(s) Kai-Wen Tu 8. Performing Organization Rept. No. 'forming Organization Name and Address Department of Mathematics University of Illinois Urbana, Illinois 6l801 10. Project/Task/Work Unit No. Thesis 11. Contract /Grant No. onsoring Organization Name and Address Graduate College Department of Computer Science University of Illinois Urbana, I llinoic 6l801 13. Type of Report & Period Covered Thesis research 14. pplementary Notes In this thesis we are concerned with variable step size multistep methods which are generalizations of I (a. y . + he. f . ) = i=n 1 n_1 1 n_1 ■y Words and Document Analysis. 17a. Descriptors Adams -Moulton-Bashf orth Interpolation Multi-step Mult i -value Variable Step Size dentif iers/Open-Ended Terms lOSATI Field/Group ailability Statement unlimited distribution 1 NTIS-35 (10-70) 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages .21 22. Price U5COMM-DC 40329-P7 1 r 5 o - i m& mm wft tniiHl HI mWBfl&mWmWBmSlR mmLwmm m H