Amorphous computing is the development of organizational principles and programming languages for obtaining coherent behavior from the cooperation of myriads of unreliable parts that are interconnected in unknown, irregular, and time-varying ways. The impetus for amorphous computing comes from developments in microfabrication and fundamental biology, each of which is the basis of a kernel technology that makes it possible to build or grow huge numbers of almost-identical information-processing units at almost no cost. This paper sets out a research agenda for realizing the potential of amorphous computing and surveys some initial progress, both in programming and in fabrication. We describe some approaches to programming amorphous systems, which are inspired by metaphors from biology and physics. We also present the basic ideas of cellular computing, an approach to constructing digital-logic circuits within living cells by representing logic levels by concentrations DNA-bin...

Abstract. Differential equations of the form ˙x = 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. 1. 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., ˙x = 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 write X = A + B in 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 ” formula [26]) ϕ: x ↦ → x ′ = exp(tA) exp(tB)(x) = x(t) + O(t 2). (1.1) The 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

We present new symmetric fourth and sixth-order symplectic Partitioned Runge-- Kutta and Runge--Kutta--Nystrom methods. We studied compositions using several extra stages, optimising the efficiency. An effective error, E f , is defined and an extensive search is carried out using the extra parameters. The new methods have smaller values of E f than other methods found in the literature. When applied to several examples they perform up to two orders of magnitude better than previously known method, which is in very good agreement with the values of E f .

Abstract. Geometric integration is the numerical integration of a differential equation, while preserving one or more of its “geometric ” properties exactly, i.e. to within round-off error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature. Geometric Integrators for ODEs 2 1.

A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler-Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge-Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-C-C* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.

This paper aims to give an introduction to the relatively new eld of geometric integration.

Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is “full”, then the order of accuracy of orbit capture (i.e. approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere. 1 Introduction. Geometric integration techniques have become increasingly popular in the modern approach to numerical

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Classical molecular dynamics simulation of a macromolecule requires the use of an efficient time-stepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on meaningful time intervals is neither obtainable nor desired, but some restrictions, such as symplecticness, can be imposed on the discretization which tend to imply good long term behavior. The presence of a variety of types and strengths of interatom potentials in standard molecular models places severe restrictions on the timestep for numerical integration used in explicit integration schemes, so much recent research has concentrated on the search for alternatives that possess (1) proper dynamical properties, and (2) a relative insensitivity to the fastest components of the dynamics. We survey several recent approaches.

We present explicit, adaptive symplectic (EASY) integrators for the numerical integration of Hamiltonian systems with greatly varying time-scales. A time regularisation is considered using the Poincare transformation. This gives a new Hamiltonian which is usually not separable, and to recover the original separability a canonical transformation is considered. A backward error analysis for the numerical integration with a splitting symplectic integrator is presented. For a one-dimensional near singular problem, this analysis reveals a strong dependence of the performance of the method with the choice of the regularisation function, g, and the order of the method. The optimal choice corresponds to the function g which nearly preserves the scaling invariance of the system. Numerical examples supporting this result are presented. Finally, an EASY method for the two- and three-dimensional Lennard-Jones problem, nearly preserving scaling invariance is also presented.