Numerical solution of hamiltonian systems in reaction-diffusion by symplectic difference schemes

Abstract

Discrete models in time and space of Fishers equation, ∂u ∂t = ∂ 2u ∂x 2 + f(u) , in reaction diffusion are numerous in mathematical biology ( Weinberger, SIAM J. Math. Anal. 13, 353 (1982) and the references therein). For f( u) = u(1− u) and no dissipation, May ( Nature 261, 459 (1976) ), using the Euler discretization of the time derivative, found stable solutions (period 2 in time) provided the time step satisfies 2 < k ⩽ √6, the linearized stability for period 1 solutions being 0 < k ⩽ 2. When the dissipation term in discretised form is added to May's ordinary difference scheme, it is shown by Grifliths and Mitchell ( IMA J. Numer. Anal. 8, 435 (1988); Numerical Analysis, Pitman Res. Notes in Math., Vol. 140, Longman, Sci. Tech., Harlow, 1986) , and Sleeman ( Proc. Roy. Soc. London Ser. A 425, 17 (1989) ) that the stable period 2 in time solutions persist. Here it is shown (Sleeman, op. cit.), that when the dissipation term in continuous form is added to May's difference equation, solutions period 2 in time for each value of x satisfy a Hamiltonian system in space. The latter, being non integrable, is solved numerically by Symplectic difference schemes constructed to maintain the values of the Hamiltonain energy up to large values of the space variable ( Feng Kang and his co-authors ( J. Comput. Math. 4, 279 (1986); Lect. Notes in Math., Vol. 1297, Springer-Verlag, New York, 1987) ). The shape of the solution, in calculations involving 200,000 space steps, is shown to depend crucially on the type and location of the fixed points of the Hamiltonian system in phase space at the position of the initial data at x=0 relative to these fixed points.