Quadratic invariants and multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

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1 Quadratc nvarants and mult-symplectcty of parttoned Runge-Kutta methods for Hamltonan PDEs Yajuan Sun Insttute of Computatonal Mathematcs and Scentfc/Engneerng Computng Academy of Mathematcs and System Scences, Chnese Academy of Scences Bejng , Chna Abstract In ths paper, we study the preservaton of quadratc conservaton laws of Runge-Kutta methods and parttoned Runge-Kutta methods for Hamltonan PDEs and establsh the relaton between mult-symplectcty of Runge-Kutta method and ts quadratc conservaton laws. For Schrödnger equatons and Drac equatons, the relaton mples that mult-sympletc Runge- Kutta methods appled to equatons wth approprate boundary condtons can preserve the global norm conservaton and the global charge conservaton respectvely. Keywords Quadratc conservaton law, Parttoned Runge-Kutta method, Mult-symplectc Hamltonan system 1 Introducton Consder the dfferental equaton z t = f(z), z R m, (1) a non-constant functon I : R m R s called ts frst ntegral, f I(z) satsfes I(z) T f(z) = 0, for all z R m. Ths mples that di dt = 0,.e., I(z(t)) s a constant along the ntegral curves of (1). A numercal scheme z k+1 = ϕ t (z k ) appled to (1) s called nvarant-preservng f the frst ntegral I s preserved exactly,.e. I(z k+1 ) = I(z k ). Preservng frst ntegrals s mportant not only because of ther The work s supported by the Natonal Scence Foundaton of Chna under grant Correspondng author. E-mal address: sunyj@lsec.cc.ac.cn (Y. J. Sun) 1

2 physcal relevance n mechancs and astronomy, but also because they can ensure the long tme computng stablty. Specally, f I(z) s a quadratc functon, then I s called a quadratc nvarant of (1). Quadratc nvarants appear very often n applcatons, the conservaton law of angular momentum n N body systems, the conservaton of total angular momentum and knetc energy of the rgd body moton etc. are the examples, we refer the readers nterested n t to [4, 7, 14, 16] and references theren for ther more applcatons. System (1) s called Hamltonan, f t s defned n an even dmensonal space R 2n and ts vector feld f : R 2n R 2n can be wrtten as f(z) = J 1 2n H(z), z R2n, (2) where J 2n s the standard 2n 2n skew-symmetrc matrx. Lnearzng (1) wth f n the form of (2) around z R 2n, we get the varatonal equaton of (1) as follows J 2n Z t = D zz H(z)Z, Z R 2n. (3) Defne a two form ω J2n wth J 2n on R 2n ω J2n (U, V ) = U T J 2n V, U, V R 2n. The fundamental geometrc property of Hamltonan equaton (1) s that t conserves symplectcty d dt ω J 2n (U, V ) = 0, (4) where U, V are any par of solutons of varatonal equaton (3). Applyng a numercal scheme to Hamltonan system (1), the numercal scheme s called symplectc f t conserves the dscrete verson of (4) d tω J2n (U, V ) = 0, (5) where d t s dscretzaton of the dervatve d dt, and {U, Z}, {V, Z} satsfy the dscrete varatonal equaton J 2n d tz = D zzh(z )Z. (6) As well known, n general, the quadratc nvarant-preservng numercal scheme appled to Hamltonan system (1) sn t necessary to be symplectc, and not all symplectc ntegrator can preserve the quadratc nvarants of Hamltonan system (1) [6, 16]. Concernng Runge-Kutta methods appled to Hamltonan system, the stablty matrx M of numercal shemes wth b a j + b j a j b b j,, j = 1,, s beng ts entres, connects the sympletcty to the preservaton of quadratc nvarants. The Runge-Kutta method s quadratc nvarant-preservng, f M s a zero matrx [4, 7, 14, 16]. Ths condton M = 0 was also proved to be a suffcent condton for Runge- Kutta methods beng symplectc [13, 14]. Under rreducble condton [7], Runge-Kutta method s symplectc and quadratc nvarant-preservng f and only f M = 0 [7, 13, 14, 16]. 2

3 The relaton between preservaton of quadratc frst ntegrals and symplectcty was frst stated n [3] wthout consderng the rreducblty of numercal scheme. For Runge-Kutta method, [3] has shown that the conservaton of quadratc frst ntegrals mples symplectcty. Further, ths dea was generalzed straghtly forward to B-seres ntegrator ncludng Runge-Kutta methods, general lnear methods etc, please see [5, 8] for more detals. We follow the dea n [3, 7] wth Runge-Kutta methods n the followng reformulaton. Rewrttng (3) as the followng equatons ( ) ( ) ( ) ( ) 0 J 2n U t 0 D zz H(z) U =, (U, V ) R 2n R 2n. (7) J 2n 0 V t D zz H(z) 0 V Notcng D zz H(z) s the Hessan matrx, t s obvous, ( ) ( ) d dt ((U T V T 0 J 2n U ) J 2n 0 V ) = 0. (8) Ths ndcates that (8) s a quadratc frst ntegral of (7). The Runge-Kutta method appled to (1) wth (2) s as follows z 1 = z 0 + t Z = z 0 + t =0 j=0 b J 1 2n H(Z ), a j J 1 2n H(Z j), = 0,, s. (9) After lnearzng (9), the correspondng varatonal equatons are u 1 = u 0 + t W = u 0 + t =0 j=0 b J 1 2n D zzh(z )W, a j J 1 2n D zzh(z j )W j, = 0,, s. (10) By observaton, we notce that (10) s exactly the same as that obtaned from applyng Runge-Kutta method (9) to varatonal equaton (3) of (1). So t s easy to know, f (9) preserves quadratc frst ntegrals of (1), then (10) wll preserve quadratc frst ntegrals of (3) and (7). Snce the quadratc conservaton law (8) of (7) mples the fundamental geometrc property of Hamltonan system (1), the preservaton of (8) by (10) reveals that Runge-Kutta method (9) s symplectc. On other hand, sympletcty mples conservaton of quadratc frst ntegrals [8]. For Hamltonan PDEs, some natural questons are: whch knd of conservaton laws of system can be preserved exactly by mult-symplectc numercal methods? Is the relaton between symplectcty of Runge-Kutta method and ts quadratc conservaton laws n Hamltonan ODEs also true for Hamltonan PDEs? These 3

4 questons are not trval and they are mportant. The study of these questons s man purpose of ths paper. Ths paper s organzed as follows. In the next secton, for the frst order PDEs we ntroduce the concept of quadratc conservaton laws whose examples can be found n the expresson of local energy conservaton, local momentum conservaton for wave equatons and local norm conservaton and local charge conservaton for Schrödnger equatons and Drac equatons [1, 17] respectvely. They are shown to be the global conservaton property when an approprate boundary condton, for nstance, perodc boundary condton s used [1, 17, 10, 11, 12]. The relaton between multsympelctcty of Runge-Kutta method for Hamltonan PDEs and ts quadratc conservaton laws s revealed n thrd secton. For Schrödnger equatons and Drac equatons, the preservaton of quadratc conservaton laws provdes the local norm conservaton and local charge conservaton. They mply the global norm conservaton and global charge conservaton presented n [10, 12] when apply the mult-sympelctc Runge-Kutta methods to these equatons wth perodc boundary condtons. In fourth secton, we dscuss the conservaton property of parttoned Runge-Kutta methods. 2 Quadratc conservaton laws We start ths secton from the followng frst order partal dfferental equaton [18] L 1 z t + L 2 z x = h(z), (11) where L 1, L 2 are two constant matrces, (t, x) R 2 are the ndependent varables and z R n s the dependent varable, h : R n R s a functon. It s clear that the mult-symplectc PDEs provded by [1, 2, 17] Mz t + Kz x = S(z) (12) wth M and K beng two constant skew-symmetrc matrces on R n, are a partcular class of type (11). We ntroduce the defnton of conservaton laws for PDEs n the form of (11) Defnton 2.1. If there exst two smooth functons F, G : R n R such that F (z) t + G(z) x = 0 (13) holds for arbtrary soluton z(x, t) of (11), then (13) s called a conservaton law of (11). Specally, f F (z) = z T L1 z and G(z) = z T L2 z are two quadratc functons wth L 1, L 2 beng symmetrc matrces, then (13) s wrtten as z T L1 z + zt L2 z = 0. (14) t x It s sad to be a quadratc conservaton law of (11). 4

5 Remark 2.2. (14) have the followng equvalent formulatons or h(z) T (L 1 1 )T L1 z + z T x ( L T 2 (L 1 1 ) T L1 + L 2 )z = 0, f L 1 s non-degenarate (15) h(z) T (L 1 2 )T L2 z + z T t ( L T 1 (L 1 2 )T L2 + L 1 )z = 0, f L 2 s non-degenarate (16) for arbtrary soluton z of PDE (11). In fact, f (14) s a quadratc conservaton law of (11) and L 1 s non-degenerate, then 0 = z T t L 1 z + z T x L 2 z = (h(z) T z T x L 2 T )(L 1 1 ) T L1 z + z T x L 2 z smlarly, when L 2 s non-degenerate, we have = h(z) T (L 1 1 ) T L1 z + z T x ( L 2 T (L 1 1 ) T L1 + L 2 )z, 0 = z T t L 1 z + z T x L 2 z = h(z) T (L 2 1 ) T L2 z + z T t ( L T 1 (L 2 1 ) T L2 + L 1 )z. We llustrate the quadratc conservaton law n Defnton 2.1 wth Schrödnger equatons and Drac equatons. Example 2.3. Schrödnger equaton ψ t + ψ xx + a ψ 2 ψ = 0. (17) Let ψ = p + q and set z = (p, q, v, w) T = (p, q, p x, q x ) T, (17) can be rewrtten n the followng formulaton [2, 9, 10, 11, 12, 17] q t v x = a(p 2 + q 2 )p, (18) p t w x = a(p 2 + q 2 )q, (19) p x = v, (20) q x = w. (21) Apparently, (18)-(21) s a form of (12) wth M = and K = Multplyng (18) by q, (19) by p, (20) by w and (21) by v, then takng ther sum, we gan a quadratc conservaton law of (18)-(21) qq t + pp t v x q + w x p + p x w q x v = 1 2 t (q2 + p 2 ) + x (wp vq) = 0. (22) 5

6 (22) s equvalent to form (14) wth L 1 = , L 2 = Example 2.4. Drac equaton ψ (1) t + ψ (2) x + mψ (1) + 2λ( ψ (2) 2 ψ (1) 2 )ψ (1) = 0, (23) ψ (2) t + ψ (1) x mψ (2) + 2λ( ψ (1) 2 ψ (2) 2 )ψ (2) = 0, (24) where m and λ are real constants, ψ = (ψ (1), ψ (2) ) T s a spnoral wave functon whch descrbes a partcle wth the spn 1 2 and ts entres ψ(1) and ψ (2) beng complex functons, descrbe the up and down states of the spn 1 2 partcle respectvely. (23)-(24) s rewrtten as the followng equatons [9, 12] by takng ψ (l) = p (l) + q (l), l = 1, 2 p (1) t + p (2) x mq (1) 2λ(p (2)2 + q (2)2 p (1)2 q (1)2 )q (1) = 0, (25) q (1) t + q (2) x + mp (1) + 2λ(p (2)2 + q (2)2 p (1)2 q (1)2 )p (1) = 0, (26) p (2) t + p (1) x + mq (2) + 2λ(p (2)2 + q (2)2 p (1)2 q (1)2 )q (2) = 0, (27) q (2) t + q x (1) mp (2) 2λ(p (2)2 + q (2)2 p (1)2 q (1)2 )p (2) = 0. (28) Ths mples that (25)-(28) s n the form of (12) wth two skew-symmetrc matrces M = and K = Multplyng (25) by p (1), (26) by q (1), (27) by p (2) and (28) by q (2), then takng ther sum, we gan the followng equalty p (1) p (1) t + p (2) p (2) t + q (1) q (1) t + q (2) q (2) t + p (2) x p (1) + q x (2) q (1) + p (1) x p (2) + q x (1) q (2) = 1 2 t (p(1)2 + p (2)2 + q (1)2 + q (2)2 ) + x (p(2) p (1) + q (2) q (1) ) = 0 (29) whch can be wrtten n the form of (14) wth two symmetrc matrces L 1 =

7 and L 2 = , z = (p(1), p (2), q (1), q (2) ). Therefore, Drac equaton has a quadratc conservaton law n the form of Defnton 2.1. Remark 2.5. We call quadratc conservaton laws (22) and (29) the local norm conservaton and local charge conservaton respectvely for that they are equvalent to the global conservaton law when takng the ntegral of (22) and (29) over spatal doman and choosng an approprate boundary condton. We consder (18)-(21) and (25)-(28) wth perodc boundary condtons. Integratng (22) and (29) over spatal doman, t reads d 1 dt 2 (q2 + p 2 )dx + (wp vq)dx = 0, x d 1 dt 2 (p(1)2 + p (2)2 + q (1)2 + q (2)2 )dx + x (p(2) p (1) + q (2) q (1) )dx = 0. Under perodc boundary condtons, the second terms of above equaltes vansh. Therefore, we obtan d ψ 2 dx = d (q 2 + p 2 )dx = 0, (30) dt dt d ( ψ (1) 2 + ψ (2) 2 )dx = d (p (1)2 + p (2)2 + q (1)2 + q (2)2 )dx = 0. (31) dt dt (30) shows the global norm conservaton for nonlnear Schrödnger equatons, (31) shows the global charge conservaton for Drac equaton. They are mportant conservatve physcal quanttes. 3 Runge-Kutta methods and conservaton laws In ths secton, we consder PDE (11) and assume that (11) has the quadratc conservaton law n the form of (14). Dscretzng (11) by a par of s stage and r stage Runge-Kutta methods n 7

8 space and tme respectvely, we obtan z 1 m z 0 m = t Z,m z 0 m = t z 1 z 0 = Z,m z 0 = b t Z,m, (32) a j t Z j,m, (33) j=1 bm x Z,m, (34) ã mn x Z,n, (35) n=1 L 1 t Z,m + L 2 x Z,m = h(z,m ). (36) Here, we use notaton Z,m z(c m, d t), z 0 m z(c m, 0), t Z,m t z(c m, d t), x Z,m x z(c m, d t), z 1 m z(c m, t), z 0 z(0, d t), z 1 z(, d t), and c m = a j. j=1 r n=1 ã mn, d = Theorem 3.1. If the coeffcents a j, ã mn, b, b m,, j = 1,, s, m, n = 1,, r of numercal scheme (32)-(36) satsfy b a j + b j a j b b j = 0,, j = 1,, s (37) bm ã mn + b n ã nm b n bm = 0, m, n = 1,, r, (38) then the ntegrator (32)-(36) preserves the dscrete quadratc conservaton law of (11) bm (zm 1 zm) 0 T L1 (zm 1 + zm) 0 + t b (z1 z0) T L2 (z0 + z1) = 0. Proof. Multplyng (32) by t b m L1 (z 1 m + z 0 m) and takng the sum for m from 1 to r, we obtan bm (zm 1 zm) 0 T L1 (zm 1 + zm) 0 = t b bm ( t Z,m ) T L1 (zm 1 + zm). 0 Usng equaltes (32) and (33), we proceed wth zm 1 + zm 0 = 2Z,m 2 t a j t Z j,m + t b t Z,m. j=1 (39) 8

9 Substtutng the above equalty nto (39), then bm (zm 1 zm) 0 T L1 (zm 1 + zm) 0 = 2 t = 2 t bm b ( t Z,m ) T L1 Z,m 2 t 2 + t 2 j=1 bm b ( t Z,m ) T L1 Z,m. j=1 b bm b j ( t Z,m ) T L1 t Z j,m bm b a j ( t Z,m ) T L1 t Z j,m In the last equalty, we have used the assumpton b a j + b j a j b b j = 0 for all, j = 1,, s. Smlarly, multplyng (34) by tb L2 (z0 + z 1 ) and takng the sum for from 1 to s, we have t b (z1 z0) T L2 (z0 + z1) = t It follows from (34) and (35) that Therefore, t z 0 + z 1 = b (z1 z0) T L2 (z0 + z1) = 2 t = 2 t bm x Z,m + 2Z,m 2 bm b ( x Z,m ) T L2 Z,m + t 2 2 t 2 n=1 bm b ( x Z,m ) T L2 Z,m. (40) b bm ( x Z,m ) T L2 (z0 + z1). (41) ã mn x Z,n. n=1 n=1 bm b ã mn ( x Z,m ) T L2 x Z,n bm bn b ( x Z,m ) T L2 x Z,n Here, the condton b m ã mn + b n ã nm b m bn = 0, m, n = 1,, r has been used for the last equalty. (42) 9

10 Combnng (40) and (42), we conclude bm (zm 1 zm) 0 T L1 (zm 1 + zm) 0 + t = 2 t The proof of ths Theorem s completed. b (z1 z0) T L2 (z0 + z1) bm b (( t Z,m ) T L1 Z,m + ( x Z,m ) T L2 Z,m ). Remark 3.2. In the proof of Theorem 3.1, the followng fact that ( t Z,m ) T L1 Z,m + ( x Z,m ) T L2 Z,m = 0 (43) holds for arbtrary Z,m, x Z,m, t Z,m satsfyng L 1 t Z,m + L 2 x Z,m = h(z,m ), has been used. Corollary 3.3. For Schrödnger equaton, n the sense of correspondng dscretzaton, the Runge- Kutta method wth coeffcents satsfyng (37) and (38) preserves the global norm conservaton N l=1 whch was mentoned and proved n [9]. to bm ( ψm,l 1 2 ψm,l 0 2 ) = 0 (44) Applyng (32)-(35) to Schrödnger equatons, equalty (40) n the proof of Theorem 3.1 s reduced (42) s wrtten as t bm ((qm) (p 1 m) 2 (qm) 0 2 (p 0 m) 2 ) = 2 t b ( (v1 v0)(q 1 + q0) + (w1 w0)(p 1 + p 0)) + t = 2 t b bm (Q,m t Q,m + P,m t P,m ) (45) b ((p 1 p 0)(w1 + w0) (q1 q0)(v 1 + v0)) b bm (W,m x P,m V,m x Q,m Q,m x V,m + P,m x W,m ), (46) 10

11 where t Z,m = ( t P,m, t Q,m, t V,m, t W,m ), x Z,m = ( x P,m, x Q,m, x V,m, x W,m ), Z,m = (P,m, Q,m, V,m, W,m ). Combnng (45) and (46), we obtan bm ((qm) (p 1 m) 2 (qm) 0 2 (p 0 m) 2 ) + t = 2 t + t + 2 t Extend (36) for Schrödnger equatons to b ( (v1 v0)(q 1 + q0) + (w1 w0)(p 1 + p 0)) b ((p 1 p 0)(w1 + w0) (q1 q0)(v 1 + v0)) b bm (Q,m t Q,m + P,m t P,m Q,m x V,m ) b bm (P,m x W,m + W,m x P,m V,m x Q,m ). (47) t Q,m x V,m = a(p,m 2 + Q,m 2 )P,m, (48) t P,m x W,m = a(p,m 2 + Q,m 2 )Q,m, (49) x P,m = V,m, (50) x Q,m = W,m. (51) By the smlar calculaton to Example 2.3 for the local norm conservaton n contnuous sense, we derve Q,m t Q,m + P,m t P,m Q,m x V,m + P,m x W,m + W,m x P,m V,m x Q,m = 0. Ths mples that bm ((p 1 m) 2 + (qm) 1 2 (p 0 m) 2 (qm) 0 2 ) + t + t b ( (v1 v0)(q 1 + q0) + (w1 w0)(p 1 + p 0)) b ((p 1 p 0)(w1 + w0) (q1 q0)(v 1 + v0)) = 0. (52) 11

12 Takng the sum of equaton (52) over all spatal grd ponts l = 1,, N, we derve N l=1 bm (((p 1 m,l )2 + (qm,l 1 )2 (p 0 m,l )2 (qm,l 0 )2 ) = t t N l=1 N l=1 b ((vl+1 v l )(q l+1 + q l ) (p l+1 p l )(w l+1 + w l )) b ((wl+1 w l )(p l+1 + p l ) + (q l+1 q l )(v l+1 + v l )). By observaton, the equalty on rght hand vanshes under perodc boundary condtons. Corollary 3.4. Smlarly, for Drac equaton, n the sense of correspondng dscretzaton, the Runge- Kutta method wth coeffcents satsfyng (37) and (38) preserves the global charge conservaton N l=1 bm ( ψ (1)1 m,l 2 + ψ (2)1 m,l 2 ψ (1)0 m,l 2 ψ (2)0 m,l 2 ) = 0. Applyng Theorem 3.1 to Drac equaton, we obtan + = t t bm ((q (1)1 m,l )2 (q (1)0 m,l )2 ) + bm ((q (2)1 m,l )2 (q (2)0 m,l )2 ) + b (p (1) b (p (2) l+1 p(1) l )(p (2) l+1 + p(2) l l+1 p(2) l )(p (1) l+1 + p(1) l bm ((p (1)1 m,l )2 (p (1)0 m,l )2 ) ) t ) t bm ((p (2)1 m,l )2 (p (2)0 m,l )2 ) b (q (1) b (q (2) l+1 q(1) l )(q (2) l+1 + q(2) l ) l+1 q(2) l )(q (1) l+1 + q(1) l ). (53) Takng the sum of equaton (53) over all spatal grd ponts l = 1,, N and mplementng the 12

13 perodc condton, we gan N + l=1 N l=1 = t t bm ((q (1)1 m,l )2 (q (1)0 m,l )2 + (p (1)1 m,l )2 (p (1)0 m,l )2 ) bm ((q (2)1 m,l )2 (q (2)0 m,l )2 + (p (2)1 m,l )2 (p (2),0 m,l ) 2 ) N l=1 N l=1 b ((p (1) b ((p (2) l+1 p(1) l )(p (2) l+1 + p(2) l l+1 p(2) l )(p (1) l+1 + p(1) l ) + (q (1) ) + (q (2) l+1 q(1) l )(q (2) l+1 + q(2) l+1 q(2) l )(q (1) l+1 + q(1) l )) l )) = 0. Theorem 3.1 has shown that for mult-symplectc PDE (12), the mult-symplectcty of Runge- Kutta method (32)-(36) under the assumpton of coeffcents of Theorem mples the preservaton of quadratc conservaton laws n the dscrete sense. We lke to pont out the preservaton of quadratc conservaton laws also mples the mult-symplectcty of Runge-Kutta method appled to (12). Takng the lnearzaton of (12) around z, we obtan ts varatonal equaton Defne two pre-symplectc structures on R n wth M and K MZ t + KZ x = D zz S(z)Z, z R n. (54) ω M (U, V ) = U T MV, ω K (U, V ) = U T KV, U, V R n, then we know that system (12) has mult-symplectc conservaton law where U, V satsfy varatonal equaton (54). A numercal dscretzaton of (12) t ω M(U, V ) + x ω K(U, V ) = 0, M,j t z,j + K x,j z,j =,j z S(z,j ), where z,j = z(x, t j ), (55) s called mult-symplectc f t conserves the dscrete mult-symplectc conservaton law,j t ω M (U,j, V,j ) + x,j ω K (U,j, V,j ) = 0, where,j t, x,j are dscretzatons of the dervatves t, x respectvely, {U,j, (, j) Z Z} and {V,j, (, j) Z Z} satsfy the followng dscrete varatonal equaton whch s obtaned by lnearzng numercal scheme (55). M,j t Z,j + K x,j Z,j = Dzz,j S(z,j )Z,j. 13

14 We rewrte varatonal equaton (54) as follows ( ) ( ) ( ) ( ) ( ) ( ) 0 M U t 0 K U x 0 D zz S(z) U + =, (56) M 0 V t K 0 V x D zz S(z) 0 V obvously, ( 54) have a quadratc conservaton law ( ) ( ) t ((U T V T 0 M U ) ) + M 0 V x ((U T V T ) t can be reduced to U T MV t + U T KV x ( 0 K K 0 ) ( U V ) ) = 0, (57) = 0, (58) where U, V are two solutons of varatonal equaton (54). Applyng (32)-(36) to mult-symplectc PDE (12) and takng the lnearzaton, t reads u 1 m u 0 m = t W,m u 0 m = t u 1 u 0 = b t W,m, (59) a j t W j,m, (60) j=1 bm x W,m, (61) W,m u 0 = ã mn x W,n, (62) n=1 M t W,m + K x W,m = D zz S(Z,m )W,m. (63) We notce that (59)-(63) can also be obtaned by dscretzng (54) wth the Runge-Kutta method (32)-(36). If Runge-Kutta method (32)-(36) preserves the quadratc conservaton laws of PDE (11) n the dscrete sense, then t also preserves the dscrete verson of quadratc conservaton law (57) when we apply (32)-(36) to (56). Ths mples that (58) holds n the dscrete sense wth the numercal soluton satsfyng the dscrete varatonal equatons (59)-(63). Hence we have the followng concluson Theorem 3.5. Applyng the Runge-Kutta method (32)-(36) whch preserves the dscrete quadratc conservaton laws of (11) to mult-symplectc PDE (12), then the resultng dscretzaton s multsymplectc (.e., t preserves the dscrete mult-symplectc conservaton law). 14

15 4 Conservaton laws for parttoned Runge-Kutta methods In the prevous secton, for PDE (11) we have dscussed the Runge-Kutta method and ts quadratc conservaton laws, n ths secton, we turn our attenton to parttoned Runge-Kutta method appled to (11). In general, such methods cannot conserve general quadratc conservaton laws n the form of Defnton 2.1, even n the case of [16]. Consder an ordnary dfferental equaton n the parttoned form ṗ = f(p, q), q = g(p, q). (64) ( ) ( ) ( ) Suppose p T q T C 1 C 2 p C2 T s the quadratc nvarant of (64), then we have C 3 q 0 = d ( ) ( ) ( ) dt ( p T q T C 1 C 2 p C2 T ) = p T t C 1 p + p T t C 2 q + qt T C2 T p + qt T C 3 q C 3 q = f(p, q) T C 1 p + f(p, q) T C 2 q + g(p, q) T C T 2 p + g(p, q) T C 3 q, where C 1, C 3 are two symmetrc matrces. The parttoned Runge-Kutta method for (64) s (65) We use notaton p 1 = p 0 + t q 1 = q 0 + t b f(p, Q ), ˆb g(p, Q ), P = p 0 + t a j f(p j, Q j ), Q = q 0 + t j=1 â j g(p j, Q j ). j=1 (66) α 1 j = b b j b a j b j a j, α 2 j = b ˆbj b â j ˆb j a j, α 3 j = ˆb ˆbj ˆb â j ˆb j â j,, j = 1,, s, then the preservaton of conservaton laws for (66) can be formulated as Proposton 4.1. For l = 1, 2, 3, f the followng condtons hold α l j = 0 or C l = 0,, j = 1,, s, (67) then the parttoned Runge-Kutta method (66) preserves quadratc nvarants n the followng form ( ) ( ) ( ) p T q T C 1 C 2 p C2 T C 3 q wth C 1, C 3 beng symmetrc matrces. 15

16 Proof. From these relatons of (66), t s easy to know p T 1 C 1 p 1 = p T 0 C 1 p 0 + t b (f(p, Q )) T C 1 P + t b P T C 1 f(p, Q ) + t 2 (b b j b a j b j a j )(f(p, Q )) T C 1 f(p j, Q j ),,j=1 smlarly, we can calculate p T 1 C 2q 1, q1 T C 3q 1. Therefore, ( ) ( ) ( ) p T 1 q1 T C 1 C 2 p ( ) ( ) ( ) 1 C2 T = p T 0 q0 T C 1 C 2 p 0 C 3 q 1 C2 T + 2 t b (f(p, Q )) T C 1 P C 3 q t ˆb P T C 2 g(p, Q ) + 2 t b (f(p, Q )) T C 2 Q + 2 t ˆb (g(p, Q )) T C 3 Q + ( t) 2 αj(f(p 1, Q )) T C 1 f(p j, Q j ) + 2( t) 2,j=1 + ( t) 2 αj(g(p 3, Q )) T C 3 g(p j, Q j ).,j=1 αj(f(p 2, Q )) T C 2 g(p j, Q j ) Under assumpton (67) of ths proposton and condton (65), ths proof s fnshed. Notcng the condton wth that the parttoned Runge-Kutta method preserves the quadratc conservaton laws, we expect the smlar result when t s appled to PDEs. Here PDE (11) s nvestgated. We express t n the followng blocked formulaton ( ) ( ) ( ) ( ) ( ) L 11 L 12 z (1) t L 21 L 22 z x (1) h (1) (z) L 13 L 14 z (2) + L t 23 L 24 z x (2) = h (2), (68) (z) where L 11, L 21 are n 1 n 1 matrces, ) L 14, L 24 are n 2 n) 2 matrces. We also wrte symmetrc matrces ( L11 L 1, L 2 n (14) as L L12 ( L21 1 = L T, 12 L L L22 2 = 13 L T, where 22 L L 11 and L 21 are n 1 n 1 symmetrc 23 matrces, L 13 and L 23 are n 2 n 2 symmetrc matrces. Therefore, (14) s rewrtten as 0 = ( ) ( L11 ) ( ) t ( z (1)T z (2)T L12 z (1) z (2) ) + ( ) ( L21 ) ( ) x ( z (1)T z (2)T L22 z (1) z (2) ) L T 12 = z (1) t L 13,j=1 L T 22 T ( L11 z (1) + L 12 z (2) ) + z (2) T ( LT 12 z (1) + L 13 z (2) ) t + z (1) x L 23 T ( L21 z (1) + L 22 z (2) ) + z x (2) T ( LT 22 z (1) + L 23 z (2) ). (69) 16

17 Now we apply the parttoned Runge-Kutta method to (78) z m (l)1 z m (l)0 = t b (l) t Z (l),m, (70) From (70)-(74), we get Further, t b(1) m ( z(1)1 Z (l),m z(l)0 m = t z (l) 1 z (l) 0 = Z (l),m z(l) 0 = j=1 n=1 a (l) j tz (l) j,m, (71) b(l) m x Z (l),m, (72) ã (l) mn x Z (l),n, (73) L 1 t Z,m + L 2 x Z,m = h(z,m ), l = 1, 2. (74) z (1)1 m + z (1)0 m = 2Z (1) z (2)1 m + z (2)0 m = 2Z (2) m z m (1)0 t = 2 t + 2 t,m + t,m + t t Z (1) b (2) t Z (2),m 2 t j=1,m 2 t j=1 ) T ( L 11 (z (1)1 m + z (1)0 m ) + L 12 (z (2)1 m + z (2)0 m )) + t b(2) m ( z(2)1 m z m (2)0 t a (1) j tz (1) j,m, a (2) j tz (2) j,m. ) T ( L T 12(z (1)1 m + z (1)0 m ) + L 13 (z (2)1 m + z (2)0 m )) b(1) m (( t Z (1),m )T L11 Z (1),m + ( tz (1),m )T L12 Z (2),m ) b(1) m (( t Z (2),m )T LT 12 Z (1),m + ( tz (2),m )T L13 Z (2),m ) + t t 2 + t 2,j=1,j=1,j=1 b(1) m ξ 1 j( t Z (1),m )T L11 t Z (1) j,m b(1) m ξ 2 j( t Z (1),m )T L12 t Z (2) j,m b(1) m ξ 3 j( t Z (2),m )T L13 t Z (2) j,m, 17

18 where we assume = b (2) (1) (2), b m = b m, = 1,, s, m = 1,, r and denote Smlarly, wth t ( z(1) ξ 1 j = j a (1) j j a (1) j, ξ 2 j = b (2) j a (2) j b (2) j a (1) j, ξ 3 j = b (2) b (2) j b (2) a (2) j b (2) j a (2) j. 1 z (1) 0 ) T ( L 21 (z (1) 1 + z (1) 0 ) + L 22 (z (2) 1 + z (2) 0 )) = 2 t + 2 t + t b (2) 1 z (2) 0 ) T ( L T 22(z (1) 1 + z (1) 0 ) + L 23 (z (2) 1 + z (2) 0 )) ( z(2) b(1) m (( x Z (1),m )T L21 Z (1),m + ( xz (1),m )T L22 Z (2),m ) b(1) m (( x Z (2),m )T LT 22 Z (1),m + ( xz (2),m )T L23 Z (2),m ) + t t 2 + t 2 m,n=1 m,n=1 m,n=1 µ 1 mn( x Z (1),m )T L21 x Z (1),n µ 2 mn( x Z (1),m )T L22 x Z (2),n µ 3 mn( x Z (2),m )T L23 x Z (2),n µ 1 mn = m n m a (1) mn n a (1) nm, µ 2 mn = m b (2) n m a (2) mn b (2) n a (1) nm, µ 3 mn = b (2) m b (2) n b (2) m a (2) mn b (2) n a (2) nm. In the same dea to Proposton 4.1, we reach the followng result Theorem 4.2. For l = 1, 2, 3, under the followng condtons ξ l j = 0 or L1l = 0,, j = 1,, s, (75) µ l mn = 0 or L2l = 0, m, n = 1,, r, (76) b(1) (2) m = b m, = b (2), = 1,, s, m = 1,, r, (77) 18

19 the parttoned Runge-Kutta method (70)-(74) preserves the dscrete conservaton law b(1) m (z (1)1 + t m z m (1)0 ) T ( L 11 (z m (1)1 + z m (1)0 ) + L 12 (z m (2)1 + z m (2)0 )) + b(2) m (z m (2)1 z m (2)0 ) T ( L T 12(z m (1)1 + z m (1)0 ) + L 13 (z m (2)1 + z m (2)0 )) (z (1) 1 z (1) 0 ) T ( L 21 (z (1) 1 + z (1) 0 ) + L 22 (z (2) 1 + z (2) 0 )) + t Remark 4.3. For parttoned system (78), b (2) (z (2) 1 z (2) 0 ) T ( L T 22(z (1) 1 + z (1) 0 ) + L 23 (z (2) 1 + z (2) 0 )) = 0. ( t Z (1),m )T L11 Z (1),m + ( tz (1),m )T L12 Z (2),m + ( tz (2),m )T LT 12 Z (1),m + ( tz (2),m )T L13 Z (2),m + ( x Z (1),m )T L21 Z (1),m + ( xz (1),m )T L22 Z (2),m + ( xz (2),m )T LT 22 Z (1),m + ( xz (2),m )T L23 Z (2),m = 0 holds for all Z (l),m, tz (l),m, xz (l),m, l = 1, 2 satsfyng ( ) ( ) ( L 11 L 12 L 13 L 14 t Z (1),m t Z (2),m + L 21 L 22 L 23 L 24 ) ( ) x Z (1),m x Z (2),m ( ) h (1) (Z,m ) = h (2). (78) (Z,m ) Remark 4.4. These condtons lsted n Theorem 4.2 mply the mult-symplectcty of parttoned Runge-Kutta methods (70)-(74) appled to mult-symplectc PDEs [9]. Therefore, we conclude that mult-symplectc parttoned Runge-Kutta method satsfyng the condtons of theorem 2.1 [9], preserves the dscrete quadratc law n the form of Theorem 4.2. Remark 4.5. The defnton 2.1 and the study of conservaton laws n ths paper can be generalzed naturally to hgher-dmensonal PDEs where n 2. L 1 z t + n L z x = 0, Acknowledgement I would lke to thank Prof. J. Hong for gvng me the questons dscussng n ths paper, and for hs useful comments. 19

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