BIT anuscript No. will be inserted by the editor Stability Ordinates of Adas Predictor-Corrector Methods Michelle L. Ghrist Jonah A. Reeger Bengt Fornberg Received: date / Accepted: date Abstract How far the stability doain of a nuerical ethod for approxiating solutions to differential equations extends along the iaginary axis indicates how useful the ethod is for approxiating solutions to wave equations; this axiu extent is tered the stability ordinate, also known as the iaginary stability boundary. It has previously been shown that exactly half of Adas-Bashforth, Adas-Moulton, and staggered Adas-Bashforth ethods have nonzero stability ordinates. In this paper, we consider two categories of Adas predictor-corrector ethods and prove that they follow a siilar pattern. In particular, if p is the order of the ethod, ABp-AMp ethods have nonzero stability ordinate only for p, 2, 5, 6, 9,,..., and ABp -AMp ethods have nonzero stability ordinates only for p3,4, 7,8,,2,... Keywords Adas ethods Linear ultistep ethods Stability ordinate Iaginary stability boundary Finite difference ethods Stability region Matheatics Subject Classification 2 65L6 65L2 65L2 65M6 65M2 Introduction When wave equations are posed as first-order systes and discretized in space to yield a syste of ordinary differential equations ODEs, the linearization of the resulting syste has a purely iaginary spectru, which corresponds to the fact that only propagation takes Support for M. Ghrist and J. Reeger provided by the United States Air Force. Support for B. Fornberg provided by NSF DMS-94647. M. Ghrist Departent of Matheatical Sciences, United States Air Force Acadey, USAF Acadey, CO 884, USA E-ail: ichelle.ghrist@usafa.edu J. Reeger B. Fornberg Departent of Applied Matheatics, Capus Box 526, University of Colorado, Boulder, CO 839, USA J. Reeger E-ail: jonah.reeger@colorado.edu B. Fornberg E-ail: fornberg@colorado.edu
2 place. Many classical nuerical ethods for ODEs have stability regions which include an interval of the for is I,iS I ] on the iaginary axis. We call the largest such value of S I the iaginary stability boundary ISB of the ODE integrator, which is also known as the stability ordinate. In the context of solving seidiscrete wave equations, one desires to use a ethod with a large ISB, which allows larger stable tie steps; ethods with zero ISB s i.e., no iaginary axis coverage in the stability doain are of no use. In this paper, we explore the question of which Adas ethods have nonzero ISB s. Adas-Bashforth AB, Adas-Moulton AM, and Adas predictor-corrector ethods are widely used ultistep ethods for approxiating solutions to first-order differential equations. These ethods are known for having lower coputational cost per iteration than equivalent-order Runge-Kutta ethods due to requiring only one new function evaluation per tie step while aintaining reasonably good accuracy and stability properties 7],, Ch. 6]. A standard -step Adas ethod for approxiating solutions to dy dt ft,y has the for t j+ y j+ y j + ptdt, t j where t k t +kh, h is the stepsize, and y yt. Here, pt is the polynoial interpolating the pointst k,y k for j + k j AB ethods or j + k j+ AM ethods. We will henceforth use j to siplify the notation. AB ethods have order p while AM ethods have order p+. In 2, Table G.3-], it was observed without proof that ABp ethods have nonzero ISB s only for orders p3,4, 7,8,,2,... and AMp ethods have nonzero ISBs only for orders p,2, 5,6, 9,,... An outline of a proof of this result for AB ethods was subsequently given in 4], with a ore detailed proof of the results for both AB and AM ethods appearing in 3]; in both papers, we also proved that staggered AB ABS ethods of order p have nonzero ISB s only for p 2, 3, 4, 7, 8,, 2,... For soe additional results on staggered ultistep ethods, see 5]. Henceforth, we will only consider nonstaggered ethods. This study revisits our previous results with a new forulation and then extends our results to Adas predictor-corrector ethods. In particular, we exaine the ethods ABp- AMp and ABp -AMp, both of which have order p. We are unaware of any other studies addressing the ISB s of such ethods for general order p. In 2, Table G.3-], it was claied that for such ethods, ost had nonzero ISB s while soe had zero ISB s. We now know that such ethods follow very siilar patterns to those of AB and AM ethods, with ABp- AMp ethods following the sae pattern as AMp ethods and ABp -AMp ethods following the sae pattern as ABp ethods. We now proceed with proving these results. 2 Preliinaries When solving the linear proble dy dt λy, the edge of a stability doain is described by the root ξ λh of ρr ξ σr when r travels around the unit circle r e iθ. Here, ρr and σr are the generating polynoials of the ethod see, e.g., p. 27 of 7]. When considering whether a stability doain has iaginary axis coverage or not, we wish to describe the behavior of the stability doain boundary near ξ. For an exact ethod, we have ξθlnr see, e.g. Theore 2. of 7], using ξ ρr. Thus an exact ethod satisfies σr ξ lnrln e iθ iθ. 2
3 A nuerical schee of order p will instead lead to ξθiθ + c p iθ p+ + d p iθ p+2 + O iθ p+3. 3 The sign of the first real ter in this expansion will dictate whether the stability doain boundary near the origin swings to the right or to the left of the iaginary axis. See Figure for an illustration coparing the stability doains of AB2 and AB3..8 a.8 b.6.6.4.4.2.2 Iξ Iξ.2.2.4.4.6.6.8.2...2 Reξ.8.2...2 Reξ Fig. Near θ, ξ iθ. Shown are portions of the boundaries of the stability regions for a AB2 and b AB3. a If the first real ter in the expansion of ξθ is negative, then the stability ordinate is. b If the first real ter in the expansion of ξθ is positive, then the stability ordinate is nonzero. For AB3, the ISB is 2 5.724. The intercepts on the real axes are and 6, respectively. 2. Henrici fors of AB and AM ethods In 6]pp. 9-95], Henrici gave a backwards differentiation representation of for AB and AM ethods. When applied to dy dy λy, an -step AB ethod can be represented by where y y + hλ γ k k y, 4 γ k k Siilarly, an -step AM ethod can be represented by y y + hλ s ds. 5 k γ k k y, 6 where γ k k s+ ds. 7 k
4 Henrici 6]p. 95] also establishes that Lea For all integers k, γ k >. γ k γ. 8 Proof Evaluating 5 directly gives γ and γ 2. For the general case, we first note that s k Then, s s+ s 2... s k k! γ k k s ds k Because the integrand is always positive for s, γ k >. Lea 2 For all integers k, γ k <. Proof Fro 8, we have k ss+s+2...s+k. k! ss+s+2...s+k. 9 k! γ k γ k γ k. Evaluating 7 directly gives γ and γ 2. For the general case, we use 9 in and siplify to find γ k k! This integrand is negative for s, so γ k <. s ss+s+2...s+k 2ds. 2.2 Exploring the exact solution Using ξ λh, the exact solution to dy dt λy is yte λt e ξt/h, where, without loss of generality, we have chosen t and yt. For an exact ethod, ξ iθ fro 2, so y n ynhe inθ. 2 An alternate way to view this equation is that we are seeking the exact solution to the relevant difference equation when following the root r which has r e iθ, which gives y n r n e iθ n e inθ. Lea 3 When y n e inθ, k y iθ k k 2 iθ+o iθ 2].
5 Proof For y n e inθ, y e iθ y, so k y e iθ k y + iθ+ 2! iθ2 + O iθ 3] k Corollary When y n e inθ, iθ k 2 iθ+o iθ 2] k iθ k k 2 iθ+o iθ 2]. k y iθ k + 2 k 2 iθ+o iθ 2]. 3 Proof For y n e inθ, k y e iθ k y, so by Lea 3, k y e iθ iθ k k 2 iθ+o iθ 2] Lea 4 When y n e inθ, iθ k +iθ+o iθ 2] k iθ 2 iθ+o 2] iθ k + 2 k 2 iθ+o iθ 2]. γ k k y + 2 iθ + O iθ 2. 4 Proof Fro 5, γ and γ 2. Using Lea 3, we find γ k k y Lea 5 When y n e inθ, γ +O γ k iθ k k 2 iθ+o iθ 2] iθ 2] + γ iθ+oiθ]+o iθ 2 + 2 iθ + O iθ 2. γ k k y + 2 iθ+o iθ 2. 5 Proof Fro 7, γ and γ 2. Using Corollary, we find γk k y γk iθk + 2 k 2 iθ+o iθ 2] γ +iθ + O + 2 iθ + O iθ 2. iθ 2] + γ iθ+oiθ]+o iθ 2
6 3 Revisiting stability ordinates for AB and AM ethods We now apply the Henrici fors of the Adas ethods to explore the stability ordinates of general AB and AM ethods. Theore AB ethods have nonzero ISBs only for orders p 3, 4, 7, 8,... Proof For AB ethods, we will show that c p > and d p < for all orders p, where c p and d p are defined by 3. The pattern for which ethods have nonzero ISBs then follows fro the powers of the iaginary unit in 3. For exaple, for p3, the first real ter in the expansion 3 is c 3 iθ 4 c 3 θ 4 >. Thus the boundary of the stability doain of AB3 swings to the right of the iaginary axis, and we have a nonzero ISB for this ethod, as seen in Figure. For p6, the first real ter in the expansion 3 is d 6 iθ 8 d 6 θ 8 < ; thus the stability doain boundary of AB6 swings to the left of the iaginary axis, and the ISB of this ethod is zero. We seek to find the values of c p and d p in the case of a general ABp ethod. We apply 2 to 4, using ξ λh to find e iθ +ξ γ k k y. 6 As, the AB ethod 4 reproduces the exact solution. Thus, using 2, we find Cobining 7 and 6 gives e iθ +iθ ξ iθ γ k k y iθ γ k k y. k γ k k y. 7 We now substitute for ξ using 3, where the order p for AB. Using Lea 3 and Lea 4, we find c iθ + + d iθ +2 + O iθ +3] + iθ 2 iθ+o 2] γ iθ + iθ 2 iθ+o 2] + γ + iθ +2 +Oiθ]+O iθ +3. Collecting like powers of iθ, we find that c γ and 2 c + d γ 2 + γ + so that d γ + 2 γ + 2 c γ + γ. 8 2 Fro Lea, we have c γ >. Using this result, 5, and 9 in 8 gives + d γ + γ 2 2+! 2+! ss+s+2 s+ 2s+ + 2] ds ss+s+2 s+ 2 + 2s ] ds.
7 Because 2 + 2s> for 2 and s, we find that d < for 2. Noting that p for AB ethods, exaining the sign of the first real ter in 3 establishes our result that AB ethods have nonzero stability ordinates only for orders p 3, 4, 7, 8,, 2,... Theore 2 AM ethods have nonzero ISBs only for orders p, 2, 5, 6, 9,,... Proof We first note that p Backwards Euler and p2 AM2 are well-known A-stable ethods and thus have nonzero stability ordinates; one can also check their expansions. AM has an expansion of ξ iθ 2 iθ2 +..., which has a positive first real ter. The expansion for AM2 contains only purely iaginary ters; this is to be expected since the stability doain boundary for AM2 consists of the entire iaginary axis. We now prove the general result for p 3. We seek to find the values of c p and d p in 3 for a general AMp ethod. We apply 2 to 6, using ξ λh to find e iθ +ξ γ k k y. 9 As, the AM ethod 6 reproduces the exact solution. Thus, using 2, we find Cobining 2 and 9 gives ξ iθ e iθ +iθ γ k k y iθ γ k k y. 2 k + γ k k y. We now substitute for ξ using 3, where the order p+ for AM. Using Corollary and Lea 5, we find c iθ +2 + d iθ +3 + O iθ +4] + iθ 2 iθ+o 2] γiθ +2 + 2 iθ+o iθ 2] + γ+iθ +3 +Oiθ]+O iθ +4. Collecting like powers of iθ, we find that c γ + and 2 c + d γ +2+ γ + 2. 2 Fro Lea 2, we have c γ+ <. Using this result, 7, and in 2 and siplifying gives d γ+2 2 γ+ s ss+s+2 s+ 2s 2] ds 2+2! Because s and 2s 2 are both negative for s and 2, we have d > and c < for AM ethods, exactly opposite the result for AB ethods. After exaining the sign of the first real ter in 3 and noting that p+ for AM ethods, we conclude that Adas-Moulton ethods have nonzero ISBs only for orders p, 2, 5, 6, 9,,...
8 4 Stability ordinates of Adas predictor-corrector ethods We now consider two different categories of Adas predictor-corrector ethods: ABp- AMp ethods and ABp -AMp ethods. We first give two exaples, AB-AM2 and AB2-AM2. For the predictor AB, we have ỹ y + h f t,y. For the predictor AB2, we have In both cases, the corrector AM2 is given by ỹ y + h 2 3 f t,y f t,y. 22 y y + h 2 f t,ỹ + f t,y. 23 We first consider AB-AM2. Using the expression for AB, substituting ft,yλy ξ h y, and letting y k r k to solve the resulting difference equation, we find that 23 becoes r+ 2 ξ+ξ+ ξ. 24 2 To find the boundary of the stability doain, we can solve follow the root ξ in 24 where r. The stability doain of this ethod is shown in Figure 2a. We can also let re iθ and do a Taylor expansion for ξθ to find that ξ iθ + 6 iθ3 8 iθ4 +... 25 Because the first real ter in this expansion is negative, AB-AM2 has a zero ISB. We next consider AB2-AM2. The analogous equation to 24 is r 2 r+ 2 ξ r+ ξ 2 3r + 2 ξ r 26 which leads to the expansion ξ iθ 2 iθ3 + 4 iθ4 +... Since the first real ter in this expansion is positive, AB2-AM2 has a nonzero ISB which is.29. The stability doain of this ethod is shown in Figure 2b. In general, fro 4, our AB predictor will take the for y P y + ξ M γ k k y 27 where M for ABp -AMp ethods and M for ABp-AMp ethods, where both ethods have order p. The general for of the AM corrector ethod is given by 6, where we are use y P instead of y after the backwards difference operation is done. This leads to y y + ξ y + ξ j j γ k k y + ξγ + γ + γ y P y γ k k y + ξ γ y P y, 28
9 a b.5.5.5.5 Iξ Iξ.5.5.5.5 2.5.5 Reξ 2.5.5 Reξ Fig. 2 Shown are the boundaries of the stability regions for a AB-AM2 and b AB2-AM2. The stability regions consist of the inside of these curves. For b, the ISB is.29. The intercept on the real axis is 2 for both ethods. where y P is given by 27 and we have used 8. Applying the exact solution 2 and 27 to 28 gives e iθ +ξ j γ k k y + ξ γ e iθ + ξ M γ k k y 29 We now use the exact AM and AB expressions 2 and 7 to substitute for the two instances of e iθ in 29 respectively. Siplifying gives ξ iθ γ k k y iθ M k γ k y + ξ γ ξ iθ k + iθ γ k k y 3 k M+ γ k k y ]. 4. ABp-AMp ethods We now consider general ABp-AMp ethods, which have order p. Theore 3 Predictor-corrector ABp-AMp ethods have nonzero ISBs only for orders p,2, 5,6, 9,,... Proof MORE!! c n a n n+! a n n n! n+! s+ jds< 3 j
and b n 2 + a n +n a n n! 2n! + a n n! d n n+2! c n an 2+ b n n! n+2! + n a n 2n! a n 2n+! an 2+ sn 2 n 2s n! 2n+2! j a n n+! s+ jds. The last integrand is nonnegative for n 2, so d n > for n 2. We can separately check that d 4. Substituting pn+ in 3 and 32 gives the coefficients c p and d p in 3 in ters of the order p. After exaining the sign of the first real ter in, we conclude that ABp-AMp ethods have nonzero ISBs only for orders p,2, 5,6, 9,,..., siilar to AMp ethods. 32 4.2 ABp -AMp ethods We now exaine general ABp -AMp ethods, which also have order p. MORE!! Theore 4 Predictor-corrector ABp -AMp ethods have nonzero ISBs only for orders p3,4, 7,8,... Proof Our general proof will require n ; we have already established that AB-AM2 has a zero ISB in 25; also see Figure 2. We now proceed with the general case for p 3. MORE! We can now find the expansion of ξ about iθ : where and d n ξθ iθ + c n iθ n+3 + d n iθ n+4 +... 33 c n b n n+2! + a n n + a n n+! 2 n+! B n n+3! c n 2 + a n n+! B n n+3! 3n2 5n+4 24 n+! + a n n+!n+2! b n 2n+2! a n3n 2 + n 2 n 2 an an 24n+! 2 n+! n+! a n b n 34 3 35. We clai that c n > and d n < for n. We first check that c 44 9 and d 864 243. We note that a n ss+...s+nds>n! for n 2; this can be shown via a straightforward proof by induction. We first use this to show that c n >. c n > b n n+2! + a n n+! b n n+2! + a n n+2!n+ n + a n 2 n+! n + n+! 2 n+ ns+s+ n j s+nds>. 36
We next show that d n <. We first note that the last ter in 35 is negative. We have d n < where 24n+3!n+ 24n+B n 2n+3n+b n 24n+3!n+ n j n+3n+2 n+3n 2 + n 2+2n a n ] 37 s+ jfn,sds Fn,s 24n+s 2 2 2n 2 + 3n 2 n 3 + 3n 2 + 5n+3 s 38 < 2n 2n+3 2 n 3 + 3n 2 + 5n+3 s, where in the last step we have evaluated the first ter at s to axiize it. Since this last expression is negative for n 2, the integrand in 37 is also negative, so d n < for ABp -AMp ethods for n 2, where pn+2. Since c n > and d n < for n, after exaining the sign of the first real ter in 33, we conclude that ABp -AMp ethods have nonzero ISBs only for orders p 3, 4, 7, 8,..., siilar to ABp ethods. 5 Discussion Since publishing 4], we have becoe aware of an earlier paper by Jeltsch 8]; soe of his theores apply to our results for AB and AM ethods. In particular, 8, Th. 5] shows that AB ethods have zero ISB s if p,5,9,... and AM ethods have zero ISB s if p3,7,,... Also, 8, Th. ] can be used to show our results for AB and AM ethods when p is odd. This theore can also be used to show that for p even, exactly every other even-ordered ethod will have a nonzero ISB for both AB and AM ethods; in addition, it establishes that when ABp has a nonzero ISB, the corresponding AMp ethod will have a zero ISB, and vice versa. However, we do not believe that the results given in 8] apply to ABS ethods or to Adas predictor-corrector ethods, which we have considered here. 6 Conclusions We have considered the question of when AB and AM ethods of general order p have nonzero stability ordinates, which corresponds to being stable when applied to discretized wave equations for sall enough stepsize. By applying Henrici s backwards differentiation forulation of the AB and AM ethods 6], we have proven that ABp-AMp ethods have nonzero stability ordinate only for p,2, 5,6, 9,,..., which atches AMp ethods. We have also shown that ABp -AMp ethods have nonzero stability ordinates only for p3,4, 7,8,,2,..., which atches ABp ethods. Acknowledgeents The authors are extreely grateful to Ernst Hairer for excellent suggestions on a previous for of this anuscript, including the use of the Henrici fors of AB and AM ethods.
2 References. Atkinson, K.: An Introduction to Nuerical Analysis. Wiley, New York 989 2. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cabridge University Press, Cabridge 996 3. Ghrist, M: High-order Finite Difference Methods for Wave Equations. Ph.D. thesis, Departent of Applied Matheatics, University of Colorado-Boulder, Boulder, CO 2 4. Ghrist, M., Fornberg, B., Driscoll, T.: Staggered tie integrators for wave equations, SIAM J. Nu. Anal., 38, 78 74 2 5. Ghrist, M., Fornberg, B.: Two results concerning the stability of staggered ultistep ethods, SIAM J. Nu. Anal., 54, 849 86 22 6. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. John Wiley & Sons, New York 962 7. Iserles, A.: Nuerical Analysis of Differential Equations. Cabridge University Press, Cabridge 996 8. Jeltsch, R.: A necessary condition for A-Stability of ultistep ultiderivative ethods, Math. Cop., 3, 739 746 976