METHODS FOR HAMILTONIAN PDEs

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1 HE MULI-SYMPLECICIY OF PARIIONED RUNGE-KUA MEHODS FOR HAMILONIAN PDEs Jialin Hong 1, Hongyu Liu 2 and Geng Sun 2 1 State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences AMSS, Chinese Academy of Sciences CAS, P.O.Box 2719, Beijing , P. R. China. 2 Institute of Mathematics, AMSS, CAS, Beijing , P. R. China. Abstract In this article we consider partitioned Runge-Kutta PRK methods for Hamiltonian partial differential equations PDEs, and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs. Key words : Partitioned Runge-Kutta method ; Multi-symplecticity ; Hamiltonian partial differential equation. AMS subject classification : 65M06, 65M99, 65N06, 65N99 1. Introduction It has been widely recognized that the symplectic integrator has the numerical superiority when applied to solving Hamiltonian ODEs. A systemic theory of symplectic integrators for Hamiltonian ODEs has been established by some authors. he Runge-Kutta methods play an important role in numerically solving differential equations see [1, 3, 4, 6, 9-19] and references therein. he symplectic condition of Runge-Kutta methods was founded independently by Lasagni, Sanz-Serna and Suris in 1988 see [6, 13, 15, 18] and references therein. he numerical analysis has been investigated and developed by some authors see [4, 6, 10, 13, 19] and references therein. Some characterizations of symplectic partitioned Runge-Kutta method, which are very useful for the construction of symplectic schemes for solving numerical Hamiltonian problems, were obtained by Sanz-Serna in [14], Sun in [16, 17] and Suris in [19], and recently discussed by Marsden and West in [11]. Reich in [12] considered Hamiltonian wave equations, and showed that the Gauss-Legendre discretization applied to the scalar wave equation and Schrödinger equation both in time and space direction, leads to a multi-symplectic integrator also see [9]. Motivated by ref. [6, 12, 14, 16, 17, 19], questions we considered are : are there any multi-symplectic partitioned Runge-Kutta methods for Hamiltonian PDEs? what is the characterization of multi-symplectic partitioned Runge-Kutta methods for the general case of Hamiltonian PDEs? he answer to the first question is obviously affirmative. he last question is closely related to the construction of higher order multi-symplectic schemes for Hamiltonian PDEs. In this article we consider the general case of Hamiltonian PDEs, and investigate the multi-symplecticity of partitioned Runge-Kutta methods, then present some conditions for multisymplectic partitioned Runge-Kutta methods. In the rest of this section we introduce some basic concepts on multi-symplectic discretization and multi-symplecticity of Hamiltonian PDEs, and give 1 his wor is supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, the NNSFC No and the Special Funds for Major State Basic Research Projects of China G his wor is supported by the Director Innovation Foundation of Institute of Mathematics and AMSS. 1

2 an extension version of Reich s result on the multi-symplecticity of Gauss-Legendre methods for the general case of Hamiltonian PDEs. In section 2 we present conditions for multi-symplecticity of partitioned Runge-Kutta methods when applied to a Hamiltonian PDE. In section 3, the multisymplecticity of partitioned Runge-Kutta methods for the wave equation is discussed. In section 4 we investigate conservative properties of energy and momentum for Runge-Kutta methods of linear Hamiltonian PDEs. In what follows we assume that all numerical methods proposed are numerically solvable, and only focus on the multi-sympleciticity of methods. Consider the Hamiltonian partial differential equation Mz t + Kz x = z Sz, x, t Ω R 2, 1 M and K are sew-symmetric matrices, and S is a real smooth function of variable z. As well-nown, some very important partial differential equations can be rewritten in this form see [2, 7, 8, 12] and references therein. he following is its multi-symplectic conservation law ωu, V t + κu, V x = 0, 2 ωu, V = U M V, κu, V = U K V, Ux, t and V x, t are solutions of the variational equation Mdz t + Kdz x = D zz Szdz. 3 In order to study the multi-symplecticity 2-preserving Runge-Kutta method, we introduce an uniform grid x j, t R 2 with mesh-length t in the t direction and mesh-length x in the x direction. he value of the function ψx, t at the mesh point x j, t is denoted by ψ j,. he equation 1, 2 and 3 can be, respectively, schemed numerically as M j, t M j, t z j, + K x j, z j, = z S j, j,, 4 j, t ω j, + x j, κ j, = 0, 5 dz j, + K x j, dz j, = Dzz j, S j, dz j,, 6 S j, = Sz j,, x j, t, ω j, =< MU j,, V j, > κ j, =< KU j,, V j, >, U j, and V j, are solutions of 6, and j, t, x j, respectively. he following definition is from [2] are discretizations of the derivatives t and x Definition 1 he numerical scheme 4 is called multi-symplectic if 5 is a discrete conservation law of 4. 2

3 o simplify notations, let the starting point x 0, t 0 =0, 0 in numerical methods proposed throughout this paper. he Runge-Kutta method for the equation 1 is the following Z m = z 0 m + t r j=1 a j t Z mj, 7 z 1 m = z 0 m + t r =1 b t Z m, 8 Z m = z 0 + x s n=1 ãmn x Z n, 9 z 1 = z 0 + x s m x Z m, 10 M t Z m + K x Z m = z SZ m, 11 notations are used as follows Z m zc m x, d t, z 0 m zc m x, 0, t Z mj t zc m x, d t, x Z mj x zc m x, d t, z 1 m zc m x, t. z 0 z0, d t, z 1 z x, d t, and c m = ã mn, d = n=1 a j. j=1 Corresponding variational equations to 7-11 respectively are dz m = dz 0 m + t r j=1 a jd t Z mj, 12 dz 1 m = dz 0 m + t r =1 b d t Z m, 13 dz m = dz 0 + x s n=1 ãmnd x Z n, 14 dz 1 = dz 0 + x s m d x Z m, 15 M t dz m + K x dz m = D zz SZ m dz m, 16 D zz SZ m is a symmetric matrix. heorem 1 If in the method 7-11 b b j b a j b j a j = 0 17 and mn m ã mn n ã nm = 0 18 hold for, j = 1, 2,, r and m, n = 1, 2,, s, then the method 7-11 is multi-symplectic with the conservation law x s m dz 1 m M dz 1 m dz 0 m M dz 0 m + t r =1 b dz 1 K dz 1 dz 0 K dz 0 = 0, 19 { dz 1 m, dz 0 m, dz 1, dz 0, dzm, d x Z m, d t Z m } and {dz 1 m, dz 0 m, dz 1, dz 0, dz m, d x Z m, d t Z m } are solutions of the variational equation

4 Proof. Let { dz 1 m, dz 0 m, dz 1, dz 0, dzm, d x Z m, d t Z m } {dz 1 m, dz 0 m, dz 1, dz 0, dz m, d x Z m, d t Z m } are solutions of the variational equation It follows from and that and dz m 1 M dzm 1 dz m 0 M dzm 0 = t r =1 b d t Z m M dz m + dz m M d t Z m + t 2 r j,=1 b b j b a j b j a j d t Z m M d t Z m = t r =1 b d t Z m M dz m + dz m M d t Z m, dz1 K dz1 dz0 K dz0 = x s m d x Z m K dz m + dz m K d x Z m. Using 16 and the symmetry of the matrix D zz SZ m produce d t Z m M dz m + dz m M d t Z m + d x Z m K dz m + dz m K d x Z m = Combining 20, 21 and 22, the proof of theorem is completed. Remar 1 his theorem can be extended to the Hamiltonian partial differential equation with varying coefficients Mxz t + Ktz x = z Sz, x, t, 23 Mx and Kt are sew-symmetric matrices and smooth in x and t respectively, Sz, x, t is a smooth real function. he following corollary is a natural extension of the result in [12] Corollary 1 If in 7-11, the method applied to both time direction and space direction is of Gauss- Legendre, then the method 7-11 is a multi-symplectic integrator. 2. Partitioned Runge-Kutta methods We consider the bloced Hamiltonian partial differential equation M1 M 0 pt K1 K M0 + 0 px p Sp, q M 2 q t K0 = K 2 q x q Sp, q, 24 M τ, K τ τ = 1, 2 are α α sew-symmetric matrices, M 0, K 0 are α α matrices, Sp, q is a smooth real function in p = p 1, p 2,, p α and q = q 1, q 2,, q α. he corresponding multi-symplectic conservation law is : ωu, V t + κu, V x = 0, 25 4

5 ωu, V = U M1 M 0 M 0 M 2 V, κu, V = U K1 K 0 K 0 K 2 V, Ux, t and V x, t are solutions of the variational equation M1 M 0 K1 K M0 dz M t K0 dz K x = D zz Szdz 26 2 and z = p 1, p 2,, p α, q 1, q 2,, q α. Now we apply partitioned Runge-Kutta method to the equation 24. M1 M 0 M 0 M 2 P m = p 0 m + t r j=1 a1 j tp mj, 27 Q m = q 0 m + t r j=1 a2 j tq mj, 28 p 1 m = p 0 m + t r =1 b1 tp m, 29 q 1 m = q 0 m + t r =1 b2 tq m, 30 P m = p 0 + x s n=1 ã1 mn x P n, 31 Q m = q 0 + x s n=1 ã2 mn x Q n, 32 p 1 = p 0 + x s 1 m x P m, 33 q1 = q0 + x s t P m K1 K + 0 t Q m K0 K 2 2 m x Q m, 34 x P m p SP = m, Q m, 35 x Q m q SP m, Q m we mae use of notations p 0 m pc m x, 0, p 0 p0, d t, q 0 m qc m x, 0, q 0 q0, d t, P m pc m x, d t, t P m p t c m x, d t, t Q m q t c m x, d t, p 1 m pc m x, t, p 1 p x, d t, q 1 m qc m x, t, q 1 q x, d t, Q m qc m x, d t, x P m p x c m x, d t, x Q m q x c m x, d t under the assumption that n=1 ã 1 mn = n=1 ã 2 mn = c m, j=1 a 1 j = j=1 a 2 j = d. 36 he corresponding system of variation equations of this method to to 6 is dp m = dp 0 m + t r j=1 a1 j d tp mj, 37 5

6 dq m = dq 0 m + t r j=1 a2 j d tq mj, 38 dp 1 m = dp 0 m + t r =1 b1 d tp m, 39 dq 1 m = dq 0 m + t r =1 b2 d tq m, 40 dp m = dp 0 + x s n=1 ã1 mnd x P n, 41 dq m = dq 0 + x s n=1 ã2 mnd x Q n, 42 dp 1 = dp 0 + x s 1 m d x P m, 43 dq1 = dq0 + x s 2 m d x Q m, 44 Md t Z m + Kd x Z m = A m dz m, 45 dpm dz m =, dq m d t P d t Z m = m, d t Q m d x P d x Z m = m, d x Q m Dpp SP A m = m, Q m D pq SP m, Q m, D qp SP m, Q m D qq SP m, Q m M1 M M = 0 K1 K M0, K = 0 M 2 K0. K 2 Obviously, A m is a symmetric matrix. Now we let {dp 1 m, dp 0 m, dp 1, dp 0, dp m, d t P m, d x P m, dq 1 m, dq 0 m, dq 1, dq 0, dq m, d t Q m, d x Q m } { dp 1 m, dp 0 m, dp 1, dp 0, dpm, d t P m, be solutions of the variational equations 37-45, and δ t ω m = dp 1 m, dq 1 m M dp 1 m dqm 1 δ x κ = dp 1, dq 1 K dp 1 d x P m, dq 1 m, dq 0 m, dq 1, dq 0, dq 1 dqm, dp 0 m, dq 0 m M dp 0, dq 0 K dp 0 d t Q m, dp 0 m dq 0 dq 0 m d x Q m }, By a straightforward calculation, we have δ t ω m = t r =1 dp m, dqm M +b 1 + t 2 r d 2 t P m, b d t Q m M j,=1 b1 +b 2 a2 j + b2 j a 2 +b 2 a1 j + b1 j a 2 +b 2 j b 1 b 2 j a 1 a1 j + b1 j a 1 j b2 j b2 j b1 a2 j b 1 d tp m b 2 d tq m, dq m dpm j b1 b1 j d t P mj M 1 d t P m b2 j d t Q mj M 2 d t Q m b1 j d t P mj M 0 d t Q m d t Q mj M 0 d t P m 48 6

7 and δ x κ = x s dp m, dqm K + 1 d x P m, 2 d x Q m K 1 2 dpm d xp m d xq m, dq m + x 2 s 1 m,n=1 m ã 1 1 mn + n ã nm m n d x P n K 1 d x P m + 2 m ã 2 2 mn + n ã nm m n d x Q n K 2 d x Q m + 2 m ã n mn + 1 m 1 n ã 2 nm 1 m ã 2 mn If for = 1, 2,, r and m = 1, 2,, s 2 m 1 2 n ã 1 n d x P n K 0 d x Q m d x Q n K 0 d x P m. nm 49 b 1 = b 2 = b, 1 2 m = m = m. 50 hen the corresponding multi-symplectic conservation law of the method to 5 is x m δ t ω m + t Consequently, in this case, it is sufficient for 51, which holds, that b δ x κ = =1 I 1 = 0 and I 2 = 0, 52 and I 2 I 1 = t 2 r j,=1 b1 +b 2 a2 j + b2 j a 2 +b 2 a1 j + b1 j a 2 +b 2 j b 1 b 2 j a m ã 2 mn + a1 j + b1 j a 1 j b2 j b2 j b1 a2 j j b1 = x 2 s 1 m,n=1 m ã 1 1 mn + n ã 1 nm 2 n ã nm m n + 2 m ã n mn + 1 m 1 n ã 2 nm 1 m ã 2 mn 2 m b1 j d t P mj M 1 d t P m b2 j d t Q mj M 2 d t Q m b1 j d t P mj M 0 d t Q m d t Q mj M 0 d t P m 1 1 m n d x P n K 1 d x P m d x Q n K 2 d x Q m n d x P n K 0 d x Q m d x Q n K 0 d x P m. 1 2 n ã 1 nm We let µ 1 j = b 1 a1 j + b1 j a 1 j b1 b1 j, µ 2 j = b 2 a2 j + b2 j a 2 j b2 b2 j, µ 3 j = b 2 a1 j + b1 j a 2 j b2 b1 j, ν 1 mn = 1 m ã 1 1 mn + n ã nm m n, ν 2 mn = 2 m ã 2 2 mn + n ã nm m n, ν 3 mn = 2 m ã 1 1 mn + n ã nm m n. hen the following result is concluded 7

8 heorem 2 In the method 27-35, suppose that 36 and 50 hold. he method is multisymplectic, with discrete multi-symplectic law 51, if one of following conditions holds 1. for τ = 1, 2, 3 µ τ j = 0, j = 1, 2, r and ν τ mn = 0 m, n = 1, 2,, s, 55 when M λ 0, K λ 0 λ = 1, 2, M 0 0 and K 0 0; 2. for τ = 1, 2, 3, µ 1 j = µ 2 j = 0, j = 1, 2, rresp. µ τ j = 0, j = 1, 2, r, 56 ν τ mn = 0 m, n = 1, 2,, sresp. ν 1 mn = ν 2 mn = 0 m, n = 1, 2,, s, 57 when M 0 = 0 resp. K 0 = 0 ; 3. for τ = 1, 2, µ τ j = 0, j = 1, 2, r and ν τ mn = 0 m, n = 1, 2,, s, when M 0 = 0 and K 0 = 0 ; 4. µ 3 j = ν 3 mn = 0, for, j = 1, 2,, r; m, n = 1, 2,, s, when M τ = K τ = 0 for τ = 1, 2. his is a typical multi-symplectic partitioned condition ; 5. for τ = 1, 2, µ 3 j = ν τ mn = 0, for, j = 1, 2,, r; m, n = 1, 2,, s, when M σ = K 0 = 0 for σ = 1, 2; 6. for τ = 1, 2, µ τ j = ν 3 mn = 0, for, j = 1, 2,, r; m, n = 1, 2,, s, when M 0 = K σ = 0 for σ = 1, 2; 7. µ 1 j = ν 3 mn = 0, for, j = 1, 2,, r; m, n = 1, 2,, s, when M 0 = M 2 = K σ = 0 for σ = 1, 2; 8. µ 3 j = ν 1 mn = 0, for, j = 1, 2,, r; m, n = 1, 2,, s, when M σ = K 0 = K 2 = 0 for σ = 1, 2. Now we give some remars. Remar 2 In heorem 2 we list only eight conditions for multi-symplecticity of partitioned Runge- Kutta method of By using I 1 = 0 and I 2 = 0, we can conclude more conditions for multisymplectic partitioned Runge-Kutta methods. his theorem can be extended naturally to the case of Hamiltonian partial differential equation with varying coefficients. Remar 3 It is trivial and apparent to extend heorem 1 and heorem 2 to the Hamiltonian partial differential equation with higher spatial dimension ι Mz t + K τ z xτ = z Sz, 58 τ=1 ι 2, M and K τ τ = 1, 2,, ι are sew-symmetric matrices, and S is a smooth function. 8

9 Remar 4 In heorem 2 the condition 1 implies a 1 j = a2 j for, j = 1, 2, r and ã1 mn = ã 2 mn for m, n = 1, 2,, s. In fact, in this case only one symplectic Runge-Kutta method is applied in each direction. Remar 5 Consider the nonlinear Schrödinger equation i ψ t + 2 ψ x 2 + ψ 2 ψ = 0, 59 Let ψx, t = ux, t + ivx, t. hen the equation 59 is read as { v t + 2 u x + u 2 + v 2 u = 0 2 u t + 2 v x + u 2 + v 2 v = We tae z = u, v, u x, v x, then the equation 6 can be rewritten as the following M = M z t + K z x = zsz, t, , K = Sz, t = 1 4 u2 + v u2 x + v 2 x he equation 61 accords with the case of condition 7 in heorem 2. hus the partitioned Runge- Kutta method can be applied to the equation 61. he numerical experiments of multisymplectic method applied to equation 59 has been given in the ref.[7,9]. Remar 6 We consider the nonlinear Dirac equation, ψ t = Aψ x + if ψ 1 2 ψ 2 2 Bψ, 62 ψ = ψ 1, ψ 2, i = 1, fs is a real function of a real variable s and matrices A and B are and respectively, ϕ = ϕ , ϕ 2 is sufficiently smooth. Let ψ j = u j + iv j j = 1, 2 and z = u 1, v 1, u 2, v 2. hen the equation 62 can be written as Mz t + Kz x = z Sz, 63 and M = , K = Sz = 1 2 F u2 1 + v 2 1 u 2 2 v 2 2, d the real smooth function F ζ satisfies that dζ F ζ = fζ he equation 63 is in the case of condition 6 in heorem 2., 9

10 Now we denote z = u 1, u 2, v 1, v 2, the equation 62 can be rewritten as ˆMẑ t + ˆKẑ x = ẑsẑ, 64 ˆM = , ˆK = he equation 64 is in the case of condition 4 in heorem Hamiltonian wave equations In this section we consider the scalar wave equation u tt = u xx G u, x, t Ω R 2, 65 G : R R is a smooth function. he investigation on symplectic integration for the equation 65 can be found in [4, 10] and references therein. Let ẑ = u, p, v, w, u t = v, u x = w, hen the equation 65 can be written as Mẑ t + Kẑ x = ẑsẑ, 66 M = K = he case 4 in heorem 2 is suitable for the equation 66. In [12], the following result is given by Reich Proposition 1 Let 65 be discretized in space and in time by a pair of Gauss-Legendre collocation methods with s, r, respectively, stages. hen the resulting discretization is a multi-symplectic integrator. A similar result is proven in [9] for the nonlinear Schrödinger equation. the multi-symplecticity of Preissman type scheme and the multi-symplectic structure for the wave equation have been discussed in [20]. Now we investigate the multi-symplecticity of partitioned Runge-Kutta method. for the equation 65 by using the multi-symplectic conservation law see [12] t du du t = x du du x. 67 he partitioned Runge-Kutta method applied to the equation 65 is 10

11 U m = u 0 + x s n=1 ã1 mn x U n, 68 W m = u 0 + x s n=1 ã2 mn x W n, 69 U m = u 0 m + t r j=1 a1 j tu mj, 70 V m = v 0 m + t r j=1 a2 j tv mj, 71 u 1 = u 0 + x s 1 m x U m, 72 w1 = w0 + x s 2 m x W m, 73 u 1 m = u 0 m + t r =1 b1 tu m, 74 v 1 m = v 0 m + t r =1 b2 tv m, 75 t U m = V m, x U m = W m, 76 t V m = x W m G U m, 77 under the assumption that r j=1 a1 j = r j=1 a2 j = d, 78 s n=1 ã1 mn = s n=1 ã2 mn = c m. 79 Here notations are in the following sense, U m uc m x, d t, t U m t uc m x, d t, x U m x uc m x, d t, u 0 u0, d t, u 1 u x, d t, u 0 m uc m x, 0, u 1 m uc m x, t. heorem 3 In the method 68-77, assume that 78, 79 and b 1 = b = b, m = m = m, m n m ã 2 2 mn n ã 1 nm = 0, 81 b 1 b2 j b 1 a2 j b2 j a 1 j = 0 82 hold for m, n = 1, 2,, s;, j = 1, 2,, s. hen the method is multi-symplectic with a discrete multi-symplectic conservation law t b du 1 dw1 du 0 dw0 = x =1 m du 1 m dvm 1 du 0 m dwm Proof. It follows from and the conditions that and On the other hand, 77 implies du 1 dw 1 du 0 dw 0 = x s m du m d x W m du 1 m dv 1 m du 0 m dv 0 m = t r =1 b du m d t V m du m d x W m = du m d t V m

12 From 84, 85 and 86, the discrete conservation law 83 is concluded. his completes the proof. Remar 7 76 and 77 imply that 78 and 79, in essence, are not necessary for the characterization of multi-symplectic partitioned Runge-Kutta methods he conservation of energy and momentum It has been shown, by S. Reich in [12] also see [2, 9], that multi-symplectic Gauss-Legendre schemes preserve both the discrete energy and momentum conservation laws exactly for linear Hamiltonian PDEs. In this section we show that the scheme 7-11 preserves the discrete energy and momentum conservation laws exactly for linear Hamiltonian PDEs Mz t + Kz x = z Sz, 87 M and K are sew-symmetric matrices, and Sz = 1 2 z Az, A is a symmetric matrix. he equation 87 has the energy conservation law t Ez + x F z = 0, 88 t Iz + x Gz = 0, 89 Ez = 1 2 z Az 1 2 xz K z, F z = 1 2 tz K z, Gz = 1 2 z Az 1 2 tz M z, Iz = 1 2 xz M z. heorem 4 Under the assumptions of heorem 1, if the matrices of RK methods in the method 7-11 are invertible, then the method 7-11 has a discrete energy conservation law x m Ezm 1 Ezm 0 + t b F z1 F z0 = 0 90 =1 and a discrete momentum conservation law x m Izm 1 Izm 0 + t b Gz1 Gz0 = =1 Proof. First of all, we introduce the system x Z m = x z 0 m + t a j t x Z mj, 92 x z 1 m = x z 0 m + t 12 j=1 b t x Z m, 93 =1

13 x z 0 m and t z 0 satisfy respectively, and t Z m = t z 0 + x x z 1 = x z 0 + x z 0 m = z x z 0 = z 0 + t ã mn x t Z n, 94 n=1 m x t Z m, 95 ã mn x z 0 n, 96 n=1 a j t z 0, 97 j=1 t x Z m tx zc m x, d t, x t Z m xt zc m x, d t. Because matrices A = a j r r and Ã = ã mn s s are invertible, we have In fact, 9, 92 and 96 imply that Z m = z 0 + z 0 m z x t Similar, 7, 94 and 97 imply that Z m = z 0 m + z 0 z x t t x Z m = x t Z m. 98 j=1 n=1 j=1 n=1 a j ã mn t x Z nj. 99 a j ã mn x t Z nj. 100 From 99 and 100, we conclude that 98 holds for m = 1, 2,, s and = 1, 2,, r. From the assumptions, we have therefore, 1 2 z 1 K t z 1 = 1 2 z 0 K t z 0 + x s 2 m Z m K x t Z m + x s 2 m x Z m K t Z m, F z 1 F z 0 x = 1 2 m Z m K x t Z m A similar but little bit tedious calculation leads to Ez 1 m Ez 0 m t = 1 2 b Z m K t x Z m 1 2 =1 101 m x Z m K t Z m. 102 b x Z m K t Z m. 103 his means that 90 holds. Analogously, we show that 91 holds. he proof is finished. Remar 8 he discrete conservation of energy and momentum for can be discussed in a similar way, but with tedious calculation. =1 13

14 5. Conclusion heorem 1 tells us that concatenating two symplectic Runge-Kutta methods probably produce a multi-symplectic integrator with order that we need. heorem 2 provides theoretically much more constructing ways of multi-symplectic integrators by using partitioned Runge- Kutta methods. For example, a multi-symplectic integrator of the wave equation can be produced by using the Lobatto IIIA-IIIB pair to discrete the equation both in time and in space directions. Acnowledgement. Authors would lie to than Professors and Doctors Elena Celledoni, Ernst Hairer, Arieh Iserles, Ying Liu, Cristian Lubich, Robert McLachlan, Hans Munthe-Kaas, Brynjulf Owren, Sebastian Reich, Gerhard Wanner and Antonella Zanna for useful discussions and their help. Jialin is very grateful to Anto and Hans, Elena and Brynjulf for their hospitality when he was visiting CAS, Norwegian Academy of Sciences. References [1] J.C.Butcher, he Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, [2] h.j.bridges and S.Reich, Multi-symplectic integrator : numerical schemes for Hamiltonian PDEs that conserve symplecticity, Physics Letters A [3] G.J.Cooper, Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal [4] D.B.Duncan, Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal [5] J.de Frutos & J.M.Sanz-Serna, Split-step spectral schemes for nonlinear Dirac systems, J. Comp. Phys [6] E.Hairer, C.Lubich and G.Wanner, Geometric Numerical Integration. Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics 31 Springer Berlin [7] J.Hong, Y.Liu, H. Munthe-Kaas and A.Zanna, On a multi-symplectic scheme for Schrödinger equations with varying coefficients, preprint [8] J.Hong and Y.Liu, Multi-symplecticity of the centred box discretizations for a class of Hamiltonian PDE s and an application to quasi periodically solitary wave of qpkdv equation, Preprint [9] A.L.Islas, D.A.Karpeev and C.M.Schober, Geometric integrators for the nonlinear Schrödinger equation, J. Comput. Phys

15 [10] R.I.McLachlan, Symplectic integration of Hamiltonian wave equations, Numer. Math [11] J.E.Marsden & M.West, Discrete mechanics and variational integrators, Acta Numericsa [12] S.Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comp. Phys , [13] J.M.Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BI [14] J.M.Sanz-Serna, he numerical integration of Hamiltonian systems. In :Computational Ordinary Differential Equations, ed. by J.R. Cash & I.Gladwell, Clarendon Press, Oxford, [15] J.M.Sanz-Serna and M.P.Calva, Numerical Hamiltonian Systems, Chapman & Hall, London, [16] G.Sun, A simple way of constructing symplectic Runge-Kutta methods, J. Comput. Math [17] G.Sun, Symplectic partitioned Runge-Kutta methods, J. Comput. Math [18] Y.B.Suris, On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems in Russian, In :Numerical Solution of Ordinary Differential Equations, ed. S.S. Filippov, Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow, 1988, [19] Y.B.Suris, Hamiltonian methods of Runge-Kutta type and their variational interpolation in Russian. Math. Model [20] Y.Wang and M.Qin, Multisymplectic structure and multisymplectic scheme for the nonlinear wave equation, Acta Math. Appl. Sinica

Multi-symplectic Runge-Kutta-Nyström Methods for Nonlinear Schrödinger Equations

Multi-symplectic Runge-Kutta-Nyström Methods for Nonlinear Schrödinger Equations Jialin Hong 1, Xiao-yan Liu and Chun Li State Key Laboratory of Scientific and Engineering Computing, Institute of Computational