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I thought that instead of the great number of precepts of which logic is composed, I would have enough with the four following ones, provided that I made a firm and unalterable resolution not to violate them even in a single instance. The first rule was never to accept anything as true unless I recognized it to be certainly and evidently such …. The second was to divide each of the difficulties which I encountered into as many parts as possible, and as might be required for an easier solution. (Descartes)We survey splitting methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods arise when a vector field can be split into a sum of two or more parts that are each simpler to integrate than the original (in a sense to be made precise). One of the main applications of splitting methods is in geometric integration, that is, the integration of vector fields that possess a certain geometric property (e.g., being Hamiltonian, or divergence-free, or possessing a symmetry or first integral) that one wants to preserve. We first survey the classification of geometric properties of dynamical systems, before considering the theory and applications of splitting in each case. Once a splitting is constructed, the pieces are composed to form the integrator; we discuss the theory of such ‘composition methods’ and summarize the best currently known methods. Finally, we survey applications from celestial mechanics, quantum mechanics, accelerator physics, molecular dynamics, and fluid dynamics, and examples from dynamical systems, biology and reaction–diffusion systems.

This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs). Given an ODE and one or more first integrals (i.e. constants of the motion) and/or Lyapunov functions, it is shown that the ODE can be rewritten as a 'linear-gradient system'. Discrete gradients are used to construct discrete approximations to the ODE which preserve the first integrals and Lyapunov functions exactly. The method applies to all Hamiltonian, Poisson and gradient systems, and also to many dissipative systems (those with a known first integral or Lyapunov function).

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