A Conservative Discretization of a Hamiltonian System

Abstract: The reaction–diffusion equation (RDE) is a natural way of describing a system where there is not only diffusion but also interaction with its surroundings. The RDE has been the topic of interest in previous papers for its usefulness of explaining pattern formations in nature. The static RDE has, in one dimension, the form of a Newton equation. We study a discretized version of this equation, with applications in biology (where a cell can be seen a natural discretization), economics, and computer simulations. In particular we consider a class of discretized Newton equations that allows for a conserved quantity. Using this we manage to find a conservative discretization of the $\Phi^4$ system.