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).

Abstract. Differential equations of the formẋ = X = A + B are considered, where the vector fields A and B can be integrated exactly, enabling numerical integration of X by composition of the flows of A and B. Various symmetric compositions are investigated for order, complexity, and reversibility. Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order. A new, more accurate way of applying the methods thus obtained to compositions of an arbitrary first-order integrator is described and tested. The determining equations are explored, and new methods up to 100 times more accurate (at constant work) than those previously known are given. Composition methods.Composition methods are particularly useful for numerically integrating differential equations when the equations have some special structure which it is advantageous to preserve. They tend to have larger local truncation errors than standard (Runge-Kutta, multistep) methods [4,5], but this defect can be more than compensated for by their superior conservation properties.Capital letters such as X will denote vector fields on some space with coordinates x, with flows exp(tX), i.e.,ẋ = X(x) ⇒ x(t) = exp(tX)(x(0)). The vector field X is given and is to be integrated numerically with fixed time step t. Composition methods apply when one can writein such a way that exp(tA), exp(tB) can both be calculated explicitly. Then the most elementary such method is the map (essentially the "Lie-Trotter" formulaThe advantage of composing exact solutions in this way is that many geometric properties of the true flow exp(tX) are preserved: group properties in particular. If X, A, and B are Hamiltonian vector fields then both exp(tX) and the map ϕ 1991 Mathematics Subject Classification. 65L05, 70-08, 58D07, 17B66.

A detailed understanding of the hormonal regulation of spermatogenesis is required for the informed assessment and management of male fertility and, conversely, for the development of safe and reversible male hormonal contraception. An approach to the study of these issues is outlined based on the use of well-defined in vivo models of gonadotropin/androgen deprivation and replacement, the quantitative assessment of germ cell number using stereological techniques, and the directed study of specific steps in spermatogenesis shown to be hormone dependent. Drawing together data from rat, monkey, and human models, we identify differences between species and formulate an overview of the hormonal regulation of spermatogenesis. There is good evidence for both separate and synergistic roles for both testosterone and follicle-stimulating hormone (FSH) in achieving quantitatively normal spermatogenesis. Based on relatively selective withdrawal and replacement studies, FSH has key roles in the progression of type A to B spermatogonia and, in synergy with testosterone, in regulating germ cell viability. Testosterone is an absolute requirement for spermatogenesis. In rats, it has been shown to promote the adhesion of round spermatids to Sertoli cells, without which they are sloughed from the epithelium and spermatid elongation fails. The release of mature elongated spermatids from the testis (spermiation) is also under FSH/testosterone control in rats. Data from monkeys and men treated with steroidal contraceptives indicate that impairment of spermiation is a key to achieving azoospermia. The contribution of 5␣-reduced androgens in the testis to the regulation of spermatogenesis is also relevant, as 5␣-reduced androgens are maintained during gonadotropin suppression and may act to maintain low levels of germ cell development. These concepts are also discussed in the context of male hormonal contraceptive development.