A straightforward three-point quadrature formula of closed type is derived that improves on Simpson's rule. Just using the additional information integrand's derivative at two endpoints we show error sixth order in grid spacing. Various bounds for are obtained to quantify more precisely errors. Applications numerical integration given. With these bounds, which generally better than usual Peano composite formulas can be applied integrands with lower derivatives.

Most methods for modeling dynamics posit just two time scales: a fast and slow scale. But many applications, including in continuum mechanics, possess wide variety of space-time scales. Often they I develop an approach to analytically model the spatially discretized advection diffusion with rigorous support changing resolved spatial grid scale by factor two. The analytic mapping from finer coarser is then iterated generate hierarchy models on multigrid across range scales, all whole...

A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson's rule. Just using the additional information integrand's derivative at two endpoints we show error sixth order in grid spacing. Various bounds for are obtained to quantify more precisely errors. Applications numerical integration given. With these bounds, which generally better than usual Peano composite formulas can be applied integrands with lower derivatives.

Journal Article THE BEHAVIOUR OF HARMONIC RESONANT STEADY SOLUTIONS TO A MODEL DIFFERENTIAL EQUATION Get access A. J. ROBERTS Department of Applied Mathematics and Theoretical Physics, University CambridgeSilver Street, Cambridge CB3 9EW Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Mechanics Mathematics, Volume 34, Issue 3, August 1981, Pages 287–310, https://doi.org/10.1093/qjmam/34.3.287 Published: 01 1981 history Received: 07 July 1980 Revision...

Consider the macroscale modelling of microscale spatiotemporal dynamics. Here we develop a new approach to ensure coarse scale discrete models preserve important self-adjoint properties fine The first part explores discretisation continuum second addresses how dynamics on lattice are mapped factor two coarser (as in multigrids). Such mapping may be iterated empower us future research explore dependent emergent phenomena. support dynamical systems, centre manifold, theory ensures that applies...

Similarity solutions play an important role in many fields of science: we consider here similarity stochastic dynamics. Important issues are not only the existence similarity, but also whether a solution is dynamically attractive, and if it is, to what particular does system evolve. By recasting class PDEs form which centre manifold theory may be applied resolve these this class. For definiteness, first example self-similarity Burgers' equation driven by some forced studied. Under suitable...

An averaged system for the slow-fast stochastic FitzHugh--Nagumo is derived in this paper. The rate of convergence probability obtained as a byproduct. Moreover deviation between original and studied. A martingale approach proves that described by Gaussian process. gives more accurate asymptotic approximation than previous work. References S. Cerrai M. Freidlin, Averaging principle class reaction--diffusion equations, to appear Probab. Th. Rel. Fields. doi:10.1007/s00440-008-0144-z R....

Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of microscale system at desired level. In this scheme, given model computed in small patches spread across space-time domain, with patch coupling conditions bridging unsimulated space. For accurate simulations, care must be taken designing conditions. Here we construct novel which preserve translational invariance, rotational self-adjoint symmetry, thus...

Recognise that people have many, possibly conflicting, aspects to their personality. We hypothesise each separate characteristic of a personality may be treated as an independent player in non-zero sum many game. This idea is applied the two person Prisoners' Dilemma introductory example. assume prisoner has ``mercenary'' well ``altruistic'' characteristic, and find all Nash equilibria internal conflict between characteristics. The hypothesis are composed more than one ``player'' explain...

The long term aim is to use modern dynamical systems theory derive discretisations of noisy, dissipative partial differential equations. As a first step we here consider small domain and apply stochastic centre manifold techniques model. approach automatically parametrises subgrid scale processes induced by spatially distributed noise. It important discretise equations carefully, as do here, because the sometimes subtle effects noise processes. In particular see how resonance effectively...