By C. Cercignani (auth.), Renato Spigler (eds.)
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Extra info for Applied and Industrial Mathematics: Venice - 1, 1989
Am . ,:. o:m .... > :",', ;- '. ~ (~ ......... B. Hamiltonian with 5-fold rotational symmetry, using HI. in (22), k = 5. Explicit symplectic composite scheme, order 2 (21). stepsize=r, total number of steps=N , number of steps per plot =M. Single orbit is computed in all cases. 56, N=50,OOO, M=5. 701, N=100,000, M=10. The separatrix network of H6 (proportional to H lO ) is not periodic but quasiperiodic and not connected, looks like a Penrose tiling with 10-fold symmetry. 56 (upper-right) we get trapped diffusion, resulting in rosette patterns with 10-fold symmetry.
H},H} (33) J whose truncations provide symplectic leap-frog schemes Zk+l_ z k-l = 21"JVH(zk) order 2: order 2m : zf+1 - zf- l = (34) (35) oH,mzk where 0H,mz is the m-term truncation of (33). For these schemes we can establish the conservation property as: If F(z) = z' Bz is a quadratic invariant under gk, then follows the" cross invariance" under the scheme (34) All the symplectic leap-frog schemes are explicit. The simplest 2nd order scheme (34) is one of the rare cases of existing methods which are symplectic and seems to be the oldest one.
O(O,y,r). 3) tends to U1 as T -> 00.
Applied and Industrial Mathematics: Venice - 1, 1989 by C. Cercignani (auth.), Renato Spigler (eds.)