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**Extra resources for Applied and Industrial Mathematics: Venice - 1, 1989**

**Sample text**

Arnold, Physica D33(1988), 21-25. I. Arnold, Dokl. Akad. Nauk. SSSR, 156(1964), 9-12. V. Z. M. Zaslavsky, et ai, Nature, 337(1989), 543-557. K. Fung, et ai, Phys. Rev. Lett. 56(1986), 2060. THE SPEED OF WAVEFORM METHODS FOR ODES l C. W. GEAR and FEN-LIEN JUANG Department of Computer Science University of fllinois at Urbana-Champaign ABSTRACT. This paper considers iterative solution techniques for ordinary differential equations and investigates the speed of convergence of various processes. The iterates are defined as solutions of a sequence of differential equations.

E. phase space R 2n , endowed with the symplectic form w(x, y) = x' Jy. H is called the Hamiltonian or energy function defining the system. It is customary to split the z E R2n into two parts z = (p, q), Pic = Zlc, qlc = Zn+lc, k = 1, ... ,n. In classical language p, q are called position and momentum vectors conjugate to each other. Then H(z) = H(p,q), and equation (3) can be written as dp dt = -Hq(p,q) (4) dq dt = Hp(p,q) The fundamental theorem on Hamiltonian formalism says that the solution z(t) of the canonical system (3) of energy H with initial value z(O) can be generated by a one-parameter group l = gk, called the phase flow of H, of symplectic operators of R2n (locally in t) such that z(t) = gkz(O) The system (3) can be written as The symplecticity of the phase flow implies the class of conservation laws of phase area of even dimensions 2m, m = 1"" ,n) for the Hamiltonian system (3), the case m = n is the Liouville's conservation law.

3 (upper part) the diffusion pattern is more wide-spread and coarse. For small ,.. 20000002 (lower-left), which, in addition, show the extreme sensitivity of the diffusion pattern to the stepsize ,... <. • ...... A.. ~... ... 7. Hamiltonian with 3-fold rotational symmetry, using Hk in (22), k = 3. Explicit symplectic composite scheme, order 2 (21). stepsize=1', total number of steps=N , number of steps per plot =M. Single orbit is computed in all cases. 04, N=10000, M=1. 7, N=25,OOO, M=5. The separatrix network of Hs (which is proportional to Hs) is a periodic lattice composed of triangles and hexagons.