Qin, M. Z., Wang, D. L., Zhang, M. Q.: Explicit symplectic difference scheme for separatable Hamiltonian system, J. Comput. Math. 9, 211-221 (1991).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....B Theta B . Then we get formally the form of (4. 26) dz dt = J Gamma1 ffi H ffi z : 4:27) We cite references on further results of Hamiltonian algorithms, including symmetry and conservation law [9;15;25;27] symplectic Runge Kutta method [24] multi stage symplectic schemes [19] , leap frog scheme [21;15] schemes which preserve energy, explicit symplectic schemes and its applications in the Arnold diffusion, chaos computation by symplectic schemes [13] Poisson transformation and its generating function theory and its application [16] applications in quantum ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

....fis ffi g ffs = e (fi 2ff)s (fi 2l 1 2ff 2l 1 )s 2l 1 e O(s 2l 2 ) Consequently, Condition (2:14) leads to that g ffs ffi g fis ffi g ffs = e s O(s 2l 2 ) i.e. it is of order 2l 1. Since it is revertible, it must be of order 2l 2. For more details, refer to [21, 33, 36, 52] 331 x3 Symplectic Algorithms A Geometric View point We consider now Hamiltonian systems on R 2n dp dt = GammaH q ; dq dt = H p ; p = 0 B B p 1 . pn 1 C C A ; q = 0 B B q 1 . q n 1 C C A ; or in compact form, dz dt = JrH(z) z = p q 2 R 2n ; J = J 2n ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

....putting q (or p) at integer times and p (or q) at half integer times. This is the first case of symplectic schemes ever to be used and rediscovered many times [1 5,17,24 26] and might even be traced back as early to Von Zeipel. For high order composite schemes based on (2.22) 2.23) see refs. [1 9,12,13,17]. For symplectic Runge Kutta schemes, see refs. 19,20,27 30] For symplectic leap frog schemes, see refs. 5, 7, 9, 11] For Hamiltonian algorithms in 1 dimensions, see refs. 1,4,5,9,10,18,20,25] For Hamiltonian algorithms on Poisson manifolds, see refs. 16,22] For applications of generating ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

....H1 269 is revertible and of order 2. The composition g 4 = g ff 2 ffi g fi 2 ffi g ff 2 gives a revertible explicit symplectic scheme of order 4 when 2ff fi = 1; 2ff 3 fi 3 = 0; i.e. ff = 1 2 Gamma 2 1=3 0; fi = 1 Gamma 2ff 0; 11) which was derived by Qin Wang Zhang [18] and Yoshida [23] etc. by different ways. Similarly, we can get high order symplectic schemes by this precedure. Example 1. Since OE(p) and (q) are nilpotent of degree 2, the classical separable Hamiltonian H(p; q) OE(p) q) is also symplectically separable. The composition of E OE and ....

.... ffi E OE ffi E =2 : p 1 = p Gamma 2 q (q) q = q OE p (p 1 ) p = p 1 Gamma 2 q (q) are symplectic and of order 2. The revertible composition g ff ffi g fi ffi g ff f ff ffi f fi ffi f ff give symplectic schemes of order 4 with the parameters (11) i.e. [18], 23] p 1 = p Gamma c 1 q (q) q 1 = q d 1 OE p (p 1 ) p 2 = p 1 Gamma c 2 q (q 1 ) q 2 = q 1 d 2 OE p (p 2 ) p 3 = p 2 Gamma c 3 q (q 2 ) q 3 = q 2 d 3 OE p (p 3 ) p = p 3 Gamma c 4 q (q 3 ) q = q 3 d 4 OE p (p) with the ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

....; with E(y; z) dy) 2 2F (y; z)dydx F (y; z) dz) 2 as the first fundamental form on the surface (in parameters y = latitude angle, z = OE = longitude angle) in question. Contact algorithm is the 2nd order e C (4.7) non contact algorithm is a 3rd order iplicit R.K. The results are due to [21]. x5 Volume preserving Algorithms for Source free Systems The source free systems on R m are governed by the m D volume structure, the underlying Lie algebra L = SVm consists of all source free vector fields a, div a = tr a = 0, the underlying Lie group G = SDm consists of all near identity ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

....= 1 Gamma 2ff 0: 2.7 ) Generally, if g s e s a is revertible and of order 2l, then the revertible composite (2.7) of g s is of order 2(l 1) when 2ff fi = 1; 2ff 2l 1 fi 2l 1 = 0; 2:8) i.e. ff = 1 2 Gamma 2 1= 2l 1) 0; fi = 1 Gamma 2ff 0: 2. 8 ) For more details, see [19, 28, 31, 44]. We give some elementary algorithms, i.e. the explicit Euler method E, the implicit Euler method I, the centered Euler method C. E, I, C, especially the oldest and simplest E, will be the basic components for the composition of structure preserving algorithms. Then transition maps for step size ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

.... for Hamiltonian systems with remarkable performance, extensive computer experimentation 200 shows decisively their overwhelming superiority over the conventional non symplectic algorithms, especially in global, structural and qualitative aspects and in long term tracking capabilities, [6, 3, 7]. The related algebraic and geometric considerations seem to promise more. Acknowledgements The author is grateful to Dr. Wang Dao liu for discussions and help in preparing the manuscript. ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

.... Hamiltonian algorithm g s of order 4 can be easily constructed by splitting the time step of size s into 3 substeps of sizes ffs; fis; ffs, g s = g ffs ffi g fis ffi g ffs with 2ff fi = 1, 2ff 3 fi 3 = 0, i.e. ff = 2 Gamma 3 p 2) Gamma1 0; fi = 1 Gamma 2ff 0, [3]. It is important to note that, although Hamiltonian algorithms emerge as the proper methodology for solving Hamiltonian sysems, they are applicable as well to non Hamilto nian systems with advantage. Hamiltonian algorithms are neutral in the sense that they give the proper amount of dissipation ....

Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

.... (u; v) ffiH ffiv (u; v) 1 C A ; J = 0 B 0 Gamma1 1 0 1 C A : 3) where u = u(x; t) v = v(x; t) are two real functions, and H(u; v) is the Hamiltonian (energy) functional, the symplectic scheme can be constructed via its time dependent generating functional Phi(W; t) Phi(u; v; t)[4, 7, 11, 12]. The following two theorems illustrate the property of the generating function Phi and its relation to the symplectic scheme for a Hamiltonian system. Due to the limitation of the length of the paper, we just list the results. The interested reader may consult the literature [4, 6, 11, 12] for ....

....v; t) 4, 7, 11, 12] The following two theorems illustrate the property of the generating function Phi and its relation to the symplectic scheme for a Hamiltonian system. Due to the limitation of the length of the paper, we just list the results. The interested reader may consult the literature [4, 6, 11, 12] for details. Definition 1 A 4 Theta 4 real matrix T is called Darboux matrix if T 0 B B B 0 GammaI I 0 1 C C C A T = 0 B B B J 0 0 GammaJ 1 C C C A ; 4) where I is the 2 Theta 2 identity. It is easy to use the following block representation for a 4 Theta 4 Darboux matrix T and ....

M. Qin, D. Wang and M. Zhang, Explicit Symplectic Difference Scheme for Separatable Hamiltonian System, J. Comput. Math., 9 (1991), 211-221.

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Qin, M. Z., Wang, D. L., Zhang, M. Q.: Explicit symplectic difference scheme for separatable Hamiltonian system, J. Comput. Math. 9, 211-221 (1991).

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Qin M Z, Wang D L, Zhang M Q. Explicit symplectic difference schemes for separable Hamiltonian systems. J Comp Math. 1991 9(3): 211--221

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