arXiv subjects

A. J. Roberts

Publications and source records attributed to A. J. Roberts.

At least 73 records · Page 4Linked to original sources

arXiv2006

General tooth boundary conditions for equation free modelling

We are developing a framework for multiscale computation which enables models at a ``microscopic'' level of description, for example Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators, to perform modelling tasks at ``macroscopic'' length scales of interest. The plan is to use the microscopic rules restricted to small "patches" of the domain, the "teeth'', using interpolation to bridge the "gaps". Here we explore general boundary conditions coupling the widely separated ``teeth'' of the microscopic simulation that achieve high order accuracy over the macroscale. We present the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. We argue that classic high-order interpolation of the macroscopic field provides the correct forcing in whatever boundary condition is required by the microsimulator. Such interpolation leads to Tooth Boundary Conditions which achieve arbitrarily high-order consistency. The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly through the eigenvalues of the scheme for selected numerical problems; and secondly using the dynamical systems approach of holistic discretisation on a general class of linear \textsc{pde}s. Analytic modelling shows that, for a wide class of microscopic systems, the subgrid fields and the effective macroscopic model are largely independent of the tooth size and the particular tooth boundary conditions. When applied to patches of microscopic simulations these tooth boundary conditions promise efficient macroscale simulation. We expect the same approach will also accurately couple patch simulations in higher spatial dimensions.

math.DS↗

arXiv2005

Resolve the multitude of microscale interactions to model stochastic partial differential equations

Constructing numerical models of noisy partial differential equations is very delicate. Our long term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a second step we here consider a small domain, representing a finite element, and apply stochastic centre manifold theory to derive a one degree of freedom model for the dynamics in the element. The approach automatically parametrises the microscale structures induced by spatially varying stochastic noise within the element. The crucial aspect of this work is that we explore how many noise processes may interact in nonlinear dynamics. We see that noise processes with coarse structure across a finite element are the significant noises for the modelling. Further, the nonlinear dynamics abstracts effectively new noise sources over the macroscopic time scales resolved by the model.

math.DS↗

arXiv2005

Use the information dimension, not the Hausdorff

Multi-fractal patterns occur widely in nature. In developing new algorithms to determine multi-fractal spectra of experimental data I am lead to the conclusion that generalised dimensions $D_q$ of order $q\leq0$, including the Hausdorff dimension, are effectively \emph{irrelevant}. The reason is that these dimensions are extraordinarily sensitive to regions of low density in the multi-fractal data. Instead, one should concentrate attention on generalised dimensions $D_q$ for $q\geq 1$, and of these the information dimension $D_1$ seems the most robustly estimated from a finite amount of data.

nlin.PS↗

arXiv2005

Accurately model the Kuramoto--Sivashinsky dynamics with holistic discretisation

We analyse the nonlinear Kuramoto--Sivashinsky equation to develop accurate discretisations modeling its dynamics on coarse grids. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing isolating internal boundaries which are later removed. Comprehensive numerical solutions and simulations show that the holistic discretisations excellently reproduce the steady states and the dynamics of the Kuramoto--Sivashinsky equation. The Kuramoto--Sivashinsky equation is used as an example to show how holistic discretisation may be successfully applied to fourth order, nonlinear, spatio-temporal dynamical systems. This novel centre manifold approach is holistic in the sense that it treats the dynamical equations as a whole, not just as the sum of separate terms.

math.DS↗

arXiv2004

Higher order accuracy in the gap-tooth scheme for large-scale solutions using microscopic simulators

We are developing a framework for multiscale computation which enables models at a ``microscopic'' level of description, for example Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators, to perform modelling tasks at the ``macroscopic'' length scales of interest. The plan is to use the microscopic rules restricted to small patches of the domain, the ``teeth'', followed by interpolation to estimate macroscopic fields in the ``gaps''. The challenge addressed here is to find general boundary conditions for the patches of microscopic simulators that appropriately connect the widely separated ``teeth'' to achieve high order accuracy over the macroscale. Here we start exploring the issues in the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. For this case analytic solutions provide comparisons. We argue that classic high-order interpolation provides patch boundary conditions which achieve arbitrarily high-order consistency in the gap-tooth scheme, and with care are numerically stable. The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly using the dynamical systems approach of holistic discretisation; and secondly through the eigenvalues of selected numerical problems. When applied to patches of microscopic simulations these patch boundary conditions should achieve efficient macroscale simulation.

math.DS↗

arXiv2004

A normal form of thin fluid film equations provides initial conditions

We use dynamical systems theory to construct the normal form of the Navier--Stokes equations for the flow of a thin layer of fluid upon a solid substrate. The normal form equations illuminate the fluid dynamics by decoupling the long-term flow of interest from the rapid viscous decay of the transient shear modes. The normal form clearly shows the centre manifold of the lubrication model and shows the result that the initial condition for the fluid thickness of the lubrication model is \emph{not} the initial physical fluid thickness, but instead is modified by the initial lateral shear flow. With these initial conditions, better forecasts will be made using the lubrication model. This dynamical systems approach will also enable similar illumination of other complicated models of dynamics.

nlin.CD↗

arXiv2004

The accurate and comprehensive model of thin fluid flows with inertia on curved substrates

Consider the 3D flow of a viscous Newtonian fluid upon a curved 2D substrate when the fluid film is thin as occurs in many draining, coating and biological flows. We derive a comprehensive model of the dynamics of the film, the model being expressed in terms of the film thickness and the average lateral velocity. Based upon centre manifold theory, we are assured that the model accurately includes the effects of the curvature of substrate, gravitational body force, fluid inertia and dissipation. The model may be used to resolve wave-like phenomena in the dynamics of viscous fluid flows over arbitrarily curved substrates such as cylinders, tubes and spheres. We briefly illustrate its use in simulating drop formation on cylindrical fibres, wave transitions, Faraday waves, viscous hydraulic jumps, and flow vortices in a compound channel. These models are the most complete models for thin film flow of a Newtonian fluid; many other thin film models can be obtained by different truncations of the dynamical equations given herein.

chao-dyn↗

arXiv2003

Shear dispersion along circular pipes is affected by bends, but the torsion of the pipe is negligible

The flow of a viscous fluid along a curving pipe of fixed radius is driven by a pressure gradient. For a generally curving pipe it is the fluid flux which is constant along the pipe and so I correct fluid flow solutions of Dean (1928) and Topakoglu (1967) which assume constant pressure gradient. When the pipe is straight, the fluid adopts the parabolic velocity profile of Poiseuille flow; the spread of any contaminant along the pipe is then described by the shear dispersion model of Taylor (1954) and its refinements by Mercer, Watt et al (1994,1996). However, two conflicting effects occur in a generally curving pipe: viscosity skews the velocity profile which enhances the shear dispersion; whereas in faster flow centrifugal effects establish secondary flows that reduce the shear dispersion. The two opposing effects cancel at a Reynolds number of about 15. Interestingly, the torsion of the pipe seems to have very little effect upon the flow or the dispersion, the curvature is by far the dominant influence. Lastly, curvature and torsion in the fluid flow significantly enhance the upstream tails of concentration profiles in qualitative agreement with observations of dispersion in river flow.

nlin.PS↗

arXiv2003

Low-dimensional modelling of a generalized Burgers equation

Burgers equation is one of the simplest nonlinear partial differential equations-it combines the basic processes of diffusion and nonlinear steepening. In some applications it is appropriate for the diffusion coefficient to be a time-dependent function. Using a Wayne's transformation and centre manifold theory, we derive l-mode and 2-mode centre manifold models of the generalised Burgers equations for bounded smooth time dependent coefficients. These modellings give some interesting extensions to existing results such as the similarity solutions using the similarity method.

math-ph↗

arXiv2003

A step towards holistic discretisation of stochastic partial differential equations

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

math.DS↗

arXiv2003

A corrected quadrature formula and applications

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

math.NA↗

arXiv2003

The prisoners may be in two minds

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

math.GM↗

arXiv2003

Initialization and the quasi-geostrophic slow manifold

Atmospheric dynamics span a range of time-scales. The projection of measured data to a slow manifold, ${\cal M}$, removes fast gravity waves from the initial state for numerical simulations of the atmosphere. We explore further the slow manifold for a simple atmospheric model introduced by Lorenz and anticipate that our results will relevant to the vastly more detailed dynamics of atmospheres and oceans. Within the dynamics of the Lorenz model, we make clear the relation between a slow manifold $\cal M$ and the ``slowest invariant manifold'' ({\SIM}), which was constructed by Lorenz in order to avoid the divergence of approximation schemes for $\cal M$. These manifolds are shown to be identical to within exponentially small terms, and so the {\SIM} in fact shares the asymptotic nature of $\cal M$. We also investigate the issue of balancing initial data in order to remove gravity waves. This is a question of how to compute an ``initialized'' point on $\cal M$ whose subsequent evolution matches that from the measured initial data that in general lie off $\cal M$. We propose a choice based on the intuitive idea that the initialization procedure should not significantly alter the forecast. Numerical results demonstrate the utility of our initialization scheme. The normal form for Lorenz' atmospheric model shows clearly how to separate the dynamics of the different atmospheric waves. However, its construction demonstrates that {\em any} initialization procedure must eventually alter the forecast--the time-scale of the divergence between the initialized and the uninitialized solutions is inevitable and is inversely proportional to the square of initial level of gravity-wave activity.

nlin.CD↗

arXiv2001

Holistically discretise the Swift-Hohenberg equation on a scale larger than its spatial pattern

I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the pattern amplitude, has the rigorous support of centre manifold theory at finite grid size $h$, and naturally incorporates physical boundaries. The results presented here for the Swift-Hohenberg equation suggest the approach will form a powerful method in computationally exploring pattern selection in general. With the aid of computer algebra, the techniques may be applied to a wide variety of equations to derive numerical models that accurately and stably capture the dynamics including the influence of possibly forced boundaries.

math.NA↗

arXiv2001

Derive boundary conditions for holistic discretisations of Burgers' equation

I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their subgrid scale interactions. Here I show how boundaries to the physical domain may be naturally incorporated into the numerical modelling of Burgers' equation. We investigate Neumann and Dirichlet boundary conditions. As well as modelling the nonlinear advection, the method naturally derives symmetric matrices with constant bandwidth to correspond to the self-adjoint diffusion operator. The techniques developed here may be used to accurately model the nonlinear evolution of quite general spatio-temporal dynamical systems on bounded domains.

math.NA↗

arXiv2001

An Accurate Lubrication Model of Contaminated Coating Flows

The levelling of short-wave irregularities on a thin film of fluid is primarily due to the action of surface tension. Surface tension gradients are often created by a number of different factor including evaporation, thermal gradients or deposition of surfactants. Lubrication theory, which ignores inertia terms in favour of viscous terms, produces a system of two nonlinear pde's for the unsteady flow of a thin viscous Newtonian fluid containing an insoluble surfactant. A complex model, which systematically includes all relevant effects to these pde's, is developed using centre manifold techniques. The benefits of using these techniques to develop accurate models are in their application. Subtle variations in parameters or assumptions are able to be catered for by including or deleting the relevant terms rather than having to redevelop these models.

math.DS↗

arXiv2001

Holistic projection of initial conditions onto a finite difference approximation

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (Archilla, 1998). However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretisation of \pde{}s such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky equation (MacKenzie, 2000). In this paper the geometric view of a centre manifold is used to provide correct initial conditions for numerical discretisations (Roberts, 1997). The derived projection of initial conditions follows from the physical processes expressed in the PDEs and so is appropriately conservative. This rational approach increases the accuracy of forecasts made with finite difference models.

math.NA↗

arXiv2000

A holistic finite difference approach models linear dynamics consistently

I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing. For example, the advection-diffusion equation is found to be stably modelled for all advection speeds and all grid spacing. The theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory. When numerically solving nonlinear partial differential equations, this approach can be used to derive systematically finite difference models which automatically have excellent characteristics. Their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems.

math.NA↗