CANONICAL RUNGE-KUTTA-NYSTRÖM METHODS OF ORDERS 5 AND 6 DANIEL I. OKUNBOR AND ROBERT D. SKEEL DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 304 W. SPRINGFIELD AVE. URBANA, ILLINOIS 680-2987 Abstract. In ths paper, we construct canoncal explct 5-stage and 7-stage Runge-Kutta- Nyström methods of orders 5 and 6, respectvely, for Hamltonan dynamcal systems. Key Words: Hamltonan systems, canoncal ntegrators, symplectc ntegrators, Runge-Kutta-Nyström methods. AMS(MOS) Subject Classfcatons: 65L05 CR Subject Classfcatons: G..7. Introducton. There has been much recent nterest n dervng for Hamltonan systems () dq dt H(q, p) =, p dp dt p) = H(q,, q hgher order numercal ntegrators whch retan the canoncal (or symplectc) property of the flow of the orgnal system. Of partcular nterest have been explct Runge- Kutta-Nyström methods(rkn) for the specal separable Hamltonan (2) H(q, p) = 2 pt M p + V (q), where q and p are vectors representng, respectvely, the postons and momenta and where M s a dagonal matrx. The functon V (q) s assocated wth the potental energy and H the total energy. Ruth[9] was the frst to publsh results about canoncal numercal ntegrators. He showed that the 2nd-order -stage leapfrog/störmer/verlet method was canoncal and dscovered a 3-stage canoncal RKN method of order 3. Ruth s work was followed by consderable research n the area of constructng hgher order canoncal ntegrators[3, 6, 0, 2,, 2]. Forest and Ruth[3] derved an explct 3-stage canoncal ntegrator of order 4. Yoshda[2] was the frst to prove the exstence of canoncal ntegrators of arbtrarly hgh order. He showed how to construct a 3 k -stage method havng order 2k + 2 usng a composton of canoncal -stage method of order 2. He derved numercally 7- and 5-stage canoncal ntegrators, respectvely of orders 6 and 8 usng a Le group approach. Low stage number s desrable because of greater convenence (such as the generaton of more closely spaced output values). Stll much s unknown about the possbltes for hgher order canoncal methods nformaton that s useful n the search for practcal methods. In ths paper we derve Supported n part by the Natonal Scence Foundaton Grant DMS 90 5533 and Department of Energy Grant DE-FG02-9ER25099.
numercally 5th-order, 5-stage RKN methods and symmetrc 6th-order, 7-stage RKN methods n secton 3. A total of four 5th-order, 5-stage RKN methods are reported. Sxteen symmetrc 6th-order, 7-stage RKN methods were obtaned, three of these are equvalent (n the sense used n [6]) to the canoncal ntegrators constructed by Yoshda[2] for general separable Hamltonans. 2. Order and Canoncal Condtons. An s-stage Runge-Kutta-Nyström method for a system wth the Hamltonan (2) s gven by (3) s y = q n + c h q n + h 2 a j f(y j ), =, 2,..., s, j= s q n+ = q n + h q n + h 2 b f(y ), q n+ = q n + h = s B f(y ). = where q n = M p n and f(q) = M V (q). The method (3) s explct f a j = 0 for j. An explct s-stage RKN method wthout redundant stages s canoncal f [6, 7, ] (4) (5) b = B ( c ), s, a j = B j (c c j ), j <. If we assume that the condtons n (4) and (5) are satsfed, we have[4] the followng order condtons for RKN methods of order 5 t : B =, t 2 : B c = 2, t 3 : B c 2 = 3, t 4 : B B j (c c j ) = 6, j< t 5 : B c 3 = 4, t 6 : B B j c (c c j ) = 8, j< t 7 : B B j (c c j )c j = 24, t 8 : j< B c 4 = 5, t 9 : B B j c 2 (c c j ) = 0, t 0 : j< j< l< B B j B l (c c j )(c c l ) = 20, t : B B j c c j (c c j ) = 30, t 2 : j< 2 B B j c 2 j(c c j ) = 60, j<
t 3 : j< l<j B B j B l (c c j )(c j c l ) = 20. The condton t 7 s redundant (see Okunbor and Skeel [6]). We use a smlar approach as n [6] to show that t 2 and t 3 are also redundant: lhs of t 2 = B B j c 2 j(c c j ) j< = B B j c 2 (c c j ) j> = B B j c 2 (c c j ) B B j c 2 (c c j ) j< j = 0 ( B B j c 3 B B j c 2 ) j j = 0 ( 4 3 2 ) = 60 = rhs of t 2, lhs of t 3 = B B j B l (c c j )(c j c l ) j< l<j = B B j B l (c c j )(c c l ) j> l< = B B j B l (c c j )(c c l ) B B j B l (c c j )(c c l ) j< l< j l< = 20 B j [ B B l (c c l )c B B l (c c l )c j ] j l< l< = 20 [ 8 B j c j B B l (c c l )] j = 20 ( 8 2 6 ) = 20 = rhs of t 3. l l< The above results llustrate the proposton of Calvo and Sanz-Serna[] that states that f two Nyström trees that are equvalent (see defnton n the Appendx), then the Φ (see Appendx) that corresponds to one can be expressed n terms of Φ s of the other and trees of lower orders. In our case, the trees, f[z 2, f] and f[f[z 2 ]] n our specal notaton (see Appendx) that result respectvely, from t 9 and t 2 are equvalent. The same s true for trees, f[f[z] 2 ] and f[f[f]] that result respectvely, from t 0 and t 3. 3. Canoncal Runge-Kutta-Nyström methods. 3.. 5th-Order 5-Stage Methods. In secton 2, we showed that t 7, t 2 and t 3 are redundant for a canoncal RKN method of 5th-order, leavng us wth 0 condtons nvolvng 0 parameters. These condtons were then solved for B and c. We resorted 3
Method B c.670808923273432060 0.6949389070793259.2243909230997538270 0.63707996769983384 0.08849558325390825 0.020557569982598005 0.959970880377059876 0.795868963457535500 0.40090379269297793385 0.306624272377778837 0.2269344247902970 0.77070344943939539384.0028475205766260 0.2456466478370674795 2 0.2042028689304553890 0.872950556657583863 0.82437756359543068463 0.33524807438366649 0.3968280450302805846 0.03827009985427366062 0.40090379269664777606 0.69883375727544694289 0.959970880342390506 0.204380365459889029 3 0.0884955827263390.0205575700048534370.224390923490252870 0.3629280032307529580.6708089233070904000 0.305086089367564804 0.396828045027482022 0.967299004637649292 0.82437756359000080586 0.866475898260552609 4 0.2042028689342899909 0.2704898443392728669.00284752079466400 0.7543583352637640775 0.2269344234432960 0.2292965505604059595 Table 5th-order 5-stage Runge-Kutta-Nyström Methods to an teratve procedure because these condtons nvolve complcated expressons n B and c. We used the subroutnes HYBRD and HYBRJ of MINPACK obtaned from Netlb for determnng the soluton. HYBRD combnes Powell s method for optmzaton, QR factorzaton and the fnte dvded dfference method for computng the Jacoban matrx. HYBRJ s the same as HYBRD except that exact Jacoban matrx s used. The ntal guesses were obtaned randomly from a Gaussan dstrbuton wth mean 0 and standard devaton. About 0,000 dfferent ntal guesses were tred and only four methods were obtaned. These four methods, obtaned usng HYBRD were used as ntal solutons for the HYBRJ program to mprove the accuracy of method coeffcents. These four methods are shown n Table. The 2-norm of resduals n all 3 order condtons s 0 3 for method, 0 4 for methods 2, 0 5 for method 3 and 4. As s obvous method 3 s the adjont of method, and 4 the adjont of 2. The adjont of a method s obtaned by nterchangng h, q n and q n, respectvely, wth h, q n+ and q n+. The slght dfferences n coeffcents, ndcates the error n these values. We speculate that these are the only methods wth real coeffcents consderng the magntude of the number of ntal guesses tred. Very recently, we also found these methods n [8]. 3.2. 6th-Order 6-Stage Methods. To construct symmetrc methods of order 6, we start wth a 6-stage RKN method wth the followng condtons B = B 6, B 2 = B 5, B 3 = B 4, c = c 6, c 2 = c 5, c 3 = c 4. 4
Wth these condtons t, t 3, t 4 and t 7 can be wrtten as t : B 4 + B 5 + B 6 = 2, t 3 : B 4 c 4 ( c 4 ) + B 5 c 5 ( c 5 ) + B 6 c 6 ( c 6 ) = 2, t 4 : B 5 c 5 (B 4 + B 5 ) + B 6 c 6 ( B 6 ) + B 4 (B 4 c 4 + B 5 c 5 ) = 5 24, t 8 : B 4 c 2 4( c 4 ) 2 + B 5 c 2 5( c 5 ) 2 + B 6 c 2 6( c 6 ) 2 = 60. The condtons t 2, t 5 and t 6 are redundant gven t, t 3, t 4 and the symmetry condtons. For detals, see [5]. The condtons t 9, t 0 and t have complcated expressons even after they have been smplfed and they are omtted here. For a symmetrc 6-stage RKN method, t turns out that t, t 3, t 4, t 8 and two of t 9, t 0, t are enough to fnd B 4, B 5, B 6, c 4, c 5 and c 6. In all, there are 3 possble sets of equatons, namely t, t 3, t 4, t 8, t 9, t 0 ; t, t 3, t 4, t 8, t 9, t ; t, t 3, t 4, t 8, t 0, t. The three sets were solved by HYBRD and all solutons obtaned from each set never satsfed the mssng equaton after tryng 000 ntal guesses. We therefore state the followng conjecture. Conjecture. There s no symmetrc 6-stage RKN method of order 6. 3.3. 6th-Order 7-Stage Methods. The negatve result above motvated us to search for symmetrc 7-stage methods of order 6. The symmetry condtons n ths case are B = B 7, B 2 = B 6, B 3 = B 5, c = c 7, c 2 = c 6, c 3 = c 5, c 4 = 2. Wth these condtons t, t 3, t 4 and t 7 can now be wrtten as t : 2 B 4 + B 5 + B 6 + B 7 = 2, t 3 : 8 B 4 + B 5 c 5 ( c 5 ) + B 6 c 6 ( c 6 ) + B 7 c 7 ( c 7 ) = 2, t 4 : 8 B2 4 + B 5 c 5 (B 4 + B 5 ) + B 6 c 6 ( B 6 ) + B 7 c 7 ( B 7 ) 2B 6 B 7 c 6 = 5 24, 5
t 8 : 6 B 4 + 2B 5 c 2 5( c 5 ) 2 + 2B 6 c 2 6( c 6 ) 2 + 2B 7 c 2 7( c 7 ) 2 = 30. The unknowns n ths case are B 4, B 5, B 6, B 7, c 5, c 6 and c 7. Ths makes t lkely that the condtons t, t 3, t 4,t 8, t 9, t 0 and t can be solved unquely for those parameters. Agan, we used the routne HYBRD. After tryng 000 dfferent ntal guesses, we obtaned 6 dfferent methods as ndcated n Tables 2(a) and 2(b). The numbers n brackets represent the 2-norms of the resduals of all twenty-three order condtons. We dd not use HYBRJ to mprove the accuracy of method coeffcents as we dd n secton 3.. The 6th-order condtons are gven n the appendx for verfcaton purpose. These methods whch nclude the methods constructed by Yoshda are 7- stage all of order 6, counterexamples to what s suggested by Calvo and Sanz-Serna[]. Acknowledgement. We thank Skp Thompson for helpng us to obtan better accuracy for the method coeffcents wth the use of HYBRJ and an exact Jacoban matrx. REFERENCES [] M. P. Calvo and J. M. Sanz-Serna. Order condtons for canoncal Runge-Kutta-Nyström methods. BIT, 32:3 42, 992. [2] P. J. Channell and J. C. Scovel. Symplectc ntegraton of Hamltonan systems. Nonlnearty, 3:23 259, 990. [3] E. Forest and R. D. Ruth. Fourth-order symplectc ntegraton. Physca D, 43:05 7, 990. [4] E. Harer, S. P. Nørsett, and G. Wanner. Solvng Ordnary Dfferental Equatons I: Non-stff Systems. Sprnger-Verlag, Berln, 987. [5] D. Okunbor. Canoncal ntegraton methods for Hamltonan systems. Ph.D. Thess, n preparaton. [6] D. Okunbor and R. D. Skeel. Explct canoncal methods for Hamltonan systems. Math. Comp., 992. to appear. [7] D. Okunbor and R. D. Skeel. An explct Runge-Kutta-Nyström method s canoncal f and only f ts adjont s explct. J. SIAM. Numer. Anal., 29(2):52 527, 992. [8] M.-Z. Qn and W.-J. Zhu. Order condtons of two knds of canoncal dfference schemes. Manuscrpt, 992. [9] R. D. Ruth. A canoncal ntegraton technque. IEEE Trans. on Nucl. Sc., NS-30(4):2669 267, 983. [0] J. M. Sanz-Serna. The numercal ntegraton of Hamltonan systems. In Proc. of IMA Conference on Comput. ODEs. J. R. Cash and I. Gladwell, eds., Oxford Unv. Press. to appear. [] Y. B. Surs. Canoncal transformatons generated by methods of Runge-Kutta type for the numercal ntegraton of the system x = U. Zh. Vychsl. Mat. Mat. Fz., 29:202 2, x 989. (n Russan). Same as U.S.S.R. Comput. Maths. Phys., 29():38-44, 989. [2] H. Yoshda. Constructon of hgher order symplectc ntegrators. Physcs Letters A, 50:262 268, 990. 6
Method (0 8 ) 2(0 ) B 4 0.269875778733640373 3.5928607443524524644 B 5 = B 3 0.92697750488589358 5.07332703974230 B 6 = B 2 0.382402005280626 2.8634777099696926750 B 7 = B 0.6877400785572907 0.9474246282892259 c 4 0.50000000000000000000 0.5000000000000000000 c 5 = c 3 0.0652086298768034024 0.683748743920897662 c 6 = c 2 0.6537376948374477890 0.65797577674397523 c 7 = c 0.055866078787376572 0.207832884808624420 Method 3(0 0 ) 4(0 3 ) B 4 0.0002428604097750724 0.0505938039496037400 B 5 = B 3 0.089385007043372004 0.97242968985250792 B 6 = B 2 0.235864224823528428 0.294238370950594204 B 7 = B 0.649554220703644 0.377769434596637888 c 5 = c 3.4353359339365500 0.740069737940588248 c 6 = c 2 0.245704835957579767 0.3525743849834280 c 7 = c 0.8896673353684493504 0.88538450585663929 Method 5(0 4 ) 6(0 4 ) B 4 0.5343327657663783988.7555920855988540 B 5 = B 3 0.2676229790370947 0.825558240449056437 B 6 = B 2 0.04928998486352428 0.9043779035360692425 B 7 = B 0.075742905744540753 0.00082423284479237473 c 5 = c 3 0.8900472383405430 0.0833993496578745364 c 6 = c 2 0.3864950009055436620 0.6092663307930040 c 7 = c 0.3227869695265602456 0.44472255246284028 Method 7(0 2 ) 8(0 2 ) B 4 0.00932286739779732.835870434825980 B 5 = B 3 0.32498572599945862848.6089378395053750 B 6 = B 2 0.88659253523663693 0.786444555870850 B 7 = B 0.00520230598297240 0.4069283824555736 c 5 = c 3 0.4845993924970722986 0.6428876387382343 c 6 = c 2.37704865746950 0.9374705594596496 c 7 = c 0.0487222754723789332 0.457836323278534425 Table 2 (a) 6th-order 7-stage Runge-Kutta-Nyström methods Appendx Appendx A. Order Sx Condtons. Usng the notaton n [4], we have that an RKN method appled to a problem of the form y = f(y) s order p f and only f (6) b Φ (t) = (7) B Φ (t) = γ(t), 7, for Nyström trees wth ρ(t) p (ρ(t) + )γ(t) for Nyström trees wth ρ(t) p
Method 9(0 8 ) 0(0 ) B 4 0.787055635604306204.35863207233272890 B 5 = B 3 0.70725865603232922.776799849672486520 B 6 = B 2 0.525460387688645804 0.2355732336789232463 B 7 = B 0.5963598355050233026 0.78453604686676 c 5 = c 3 0.93726736774764095 0.568753682508557554 c 6 = c 2 0.228363909397727225 0.09769978284560674493 c 7 = c 0.7380405290506590652 0.607743947587824335 Method (0 4 ) 2(0 0 ) B 4 2.3763527443077595470 2.38944778326077997980 B 5 = B 3 2.32285222004507250 0.0052886228323448038 B 6 = B 2 0.0042606887079227608 2.4403536369698820 B 7 = B.439848679767824590.4477825624033375790 c 5 = c 3 0.62203376535070.6954883227598027000 c 6 = c 2 0.449785089209823 0.62423509575464408 c 7 = c 0.28007596060846049 0.27608788022885876 Method 3(0 2 ) 4(0 6 ) B 4 0.38064590970953586.72402906097547350 B 5 = B 3 0.68937486280925274 0.0008627046296579 B 6 = B 2 0.37962424274462893 0.827895409997737297 B 7 = B 0.0006600692650939825 0.9025528944608484900 c 5 = c 3 0.8439942572808029845.4089836623592965580 c 6 = c 2 0.8256584058543383652 0.082850893248825795 c 7 = c 2.03087978988764720 0.6000457532587368429 Method 5(0 3 ) 6(0 2 ) B 4.994884963358892820 0.4037333353894750508 B 5 = B 3 0.857347866656658757 0.008838400479785005 B 6 = B 2 0.0045588846048774229 0.69325224877087697 B 7 = B 0.90945058234089023426 0.40382993023042478752 c 5 = c 3 0.08589899029079409324.35603666090087950 c 6 = c 2.299450796360908980 0.835509096524756460 c 7 = c 0.5938356292552832625 0.84258275243476887 Table 2 (b) 6th-order 7-stage Runge-Kutta-Nyström methods where ρ(t) s the order of the tree, γ(t) s the densty of the tree and Φ (t) corresponds to the elementary weght of the Nyström tree. If condtons (4) combne wth (7) then condtons (6) are superfluous. Therefore, we concentrate on the condtons (7). Frst, we let z = y and then use a specal notaton (smlar to what was used n our Mathematca program) to represent the trees or the elementary dfferentals. We gve here the correspondence between the elementary dfferentals n our notaton and the Nyström trees for a few of the elementary dfferentals. The fat node correspond to dervatve of z and the meagre node, to the dervatve of the y. The bottom node s the root of the tree. Two Nyström trees are equvalent n the sense defned n Calvo 8
and Sanz-Serna[], f they have equal number of fat nodes, equal number of meagre nodes and dentcal branches and dffers only n ther roots. The order 6 condtons are gven n Table 3, where α(t) s the weght of the elementary dfferental. FIGURE MISSING t ρ(t) α(t) γ(t) Φ (t) f[z 5 ] 6 6 c 5 f[f 2, z] 6 5 24 j k a ja k c f[f, z 3 ] 6 0 2 j a jc 3 f[z, f[f]] 6 5 44 j k a ja jk c f[f, f[z]] 6 0 72 j k a ja k c k f[z 2, f[z]] 6 0 36 j a jc 2 c j f[f[f, z]] 6 3 240 j k a ja jk c j f[z, f[z 2 ]] 6 5 72 j a jc c 2 j f[f[z 3 ]] 6 20 j a jc 3 j f[f[f[z]]] 6 720 j k a ja jk c k Table 3 Order Sx Condtons 9