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A. J. Roberts

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

At least 91 records · Page 5Linked to original sources
arXiv2000

Holistic finite differences accurately model the dynamics of the Kuramoto-Sivashinsky equation

We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing insulating internal boundaries which are later removed. The Kuramoto-Sivashinsky equation is used as an example to show how holistic finite differences may be 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.NA↗
arXiv2000

A flexible error estimate for the application of centre manifold theory

In applications of centre manifold theory we need more flexible error estimates than that provided by, for example, the Approximation Theorem~3 by Carr (1981,1983). Here we extend the theory to cover the case where the order of approximation in parameters and that in dynamical variables may be completely different. This allows, for example, the effective evaluation of low-dimensional dynamical models at finite parameter values.

math.DS↗
arXiv1999

Fast and accurate multigrid solution of Poissons equation using diagonally oriented grids

We solve Poisson's equation using new multigrid algorithms that converge rapidly. The novel feature of the 2D and 3D algorithms are the use of extra diagonal grids in the multigrid hierarchy for a much richer and effective communication between the levels of the multigrid. Numerical experiments solving Poisson's equation in the unit square and unit cube show simple versions of the proposed algorithms are up to twice as fast as correspondingly simple multigrid iterations on a conventional hierarchy of grids.

math.NA↗
arXiv1999

Modelling the dynamics of turbulent floods

Consider the dynamics of turbulent flow in rivers, estuaries and floods. Based on the widely used k-epsilon model for turbulence, we use the techniques of centre manifold theory to derive dynamical models for the evolution of the water depth and of vertically averaged flow velocity and turbulent parameters. This new model for the shallow water dynamics of turbulent flow: resolves the vertical structure of the flow and the turbulence; includes interaction between turbulence and long waves; and gives a rational alternative to classical models for turbulent environmental flows.

patt-sol↗
arXiv1999

Holistic finite differences ensure fidelity to Burger's equation

I analyse a generalised Burger's equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the dynamics and may be constructed systematically. The trick to the application of centre manifold theory is to divide the physical domain into small elements by introducing insulating internal boundaries which are later removed. Burger's equation is used as an example to show how the concepts work in practise. The resulting finite difference models are shown to be significantly more accurate than conventional discretisations, particularly for highly nonlinear dynamics. This centre manifold approach treats the dynamical equations as a whole, not just as the sum of separate terms---it is holistic. The techniques developed here may be used to accurately model the nonlinear evolution of quite general spatio-temporal dynamical systems.

chao-dyn↗
arXiv1999

Advection-dispersion in symmetric field-flow fractionation channels

We model the evolution of the concentration field of macromolecules in a symmetric field-flow fractionation (FFF) channel by a one-dimensional advection-diffusion equation. The coefficients are precisely determined from the fluid dynamics. This model gives quantitative predictions of the time of elution of the molecules and the width in time of the concentration pulse. The model is rigorously supported by centre manifold theory. Errors of the derived model are quantified for improved predictions if necessary. The advection-diffusion equation is used to find that the optimal condition in a symmetric FFF for the separation of two species of molecules with similar diffusivities involves a high rate of cross-flow.

chao-dyn↗
arXiv1999

Computer algebra derives correct initial conditions for low-dimensional dynamical models

To ease analysis and simulation we make low-dimensional models of complicated dynamical systems. Centre manifold theory provides a systematic basis for the reduction of dimensionality from some detailed dynamical prescription down to a relatively simple model. An initial condition for the detailed dynamics also has to be projected onto the low-dimensional model, but has received scant attention. Herein, based upon the reduction algorithm in~\cite{Roberts96a}, I develop a straightforward algorithm for the computer algebra derivation of this projection. The method is systematic and is based upon the geometric picture underlying centre manifold theory. The method is applied to examples of a pitchfork and a Hopf bifurcation. There is a close relationship between this projection of initial conditions and the correct projection of forcing onto a model. I reaffirm this connection and show how the effects of forcing, both interior and from the boundary, should be properly included in a dynamical model.

chao-dyn↗
arXiv1998

Proper initial conditions for the lubrication model of the flow of a thin film of fluid

A lubrication model describes the dynamics of a thin layer of fluid spreading over a solid substrate. But to make forecasts we need to supply correct initial conditions to the model. Remarkably, the initial fluid thickness is not the correct initial thickness for the lubrication model. Theory recently developed in \cite{Roberts89b,Roberts97b} provides the correct projection of initial conditions onto a model of a dynamical system. The correct projection is determined by requiring that the model's solution exponentially quickly approaches that of the actual fluid dynamics. For lubrication we show that although the initial free surface shape contributes the most to the model's initial conditions, the initial velocity field is also an influence. The projection also gives a rationale for incorporating miscellaneous small forcing effects into the lubrication model; gravitational forcing is given as one example.

chao-dyn↗
arXiv1998

Similarity, attraction and initial conditions in an example of nonlinear diffusion

Similarity solutions play an important role in many fields of science. The recent book of Barenblatt (1996) discusses many examples. Often, outstanding unresolved issues are whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting the dynamic problem in a form to which centre manifold theory may be applied, based upon a transformation by Wayne (1997), we may resolve these issues in many cases. For definiteness we illustrate the principles by discussing the application of centre manifold theory to a particular nonlinear diffusion problem arising in filtration. Theory constructs the similarity solution, confirms its relevance, and determines the correct solution for any compact initial condition. The techniques and results we discuss are applicable to a wide range of similarity problems.

chao-dyn↗
arXiv1998

The Accurate Modelling of Thin 3D Fluid Flows with Inertia on Curved Substrates

Models and simulations of the flow of thin films of fluids have many important applications in industrial and natural processes. We consider the motion of a thin layer of an incompressible, Newtonian fluid over an arbitrary solid, stationary curved substrate. Earlier work by Roy, et al has modeled slow fluid flows in which inertia is negligible. However, in many applications a model expressed in terms of both the fluid layer thickness and the average lateral velocity is needed to resolve faster dynamics. Here we use centre manifold techniques to derive such a non-dimensional, approximate model of the three dimensional flow.

patt-sol↗
arXiv1997

Low-Dimensional Modelling of Dynamical Systems

Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is interesting. In a wide variety of situations, simple approximate models are needed to perform practical simulations and make forecasts. I review the derivation, from a mathematical description of the detailed dynamics, of accurate, complete and useful low-dimensional models of the interesting dynamics in a system. The development of centre manifold theory and associated techniques puts this modelling process on a firm basis. As in Guckenheinmer & Holmes (1983,S2.5): "... these new methods will really be conventional perturbation style analyses interpreted geometrically..." But the geometric viewpoint of dynamical systems theory greatly enriches our approach by providing a rationale for also deriving correct initial conditions, forcing and boundary conditions for the models-all essential elements of a model.

chao-dyn↗
arXiv1997

A Lubrication Model of Coating Flows over a Curved Substrate in Space

Consider the three-dimensional flow of a viscous Newtonian fluid upon an abitrarily curved substrate when the fluid film is thin as occurs in many draining, coating and biological flows. We derive a model of the dynamics of the film, the model being expressed in terms of the film thickness and the curvature tensor of the substrate. The model accurately includes the effects of the curvature of the substrate, via a physical multiple-scale approach, and gravity and inertia, via more rigorous centre manifold techniques. Numerical simulations exhibit some generic features of the dynamics of such thin fluid films on substrates with complex curvature.

patt-sol↗
arXiv1997

On the low-dimensional modelling of Stratonovich stochastic differential equations

We develop further ideas on how to construct low-dimensional models of stochastic dynamical systems. The aim is to derive a consistent and accurate model from the originally high-dimensional system. This is done with the support of centre manifold theory and techniques. Aspects of several previous approaches are combined and extended: adiabatic elimination has previously been used, but centre manifold techniques simplify the noise by removing memory effects, and with less algebraic effort than normal forms; analysis of associated Fokker-Plank equations replace nonlinearly generated noise processes by their long-term equivalent white noise. The ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations.

chao-dyn↗
arXiv1996

Low-Dimensional Modelling of Dynamics via Computer Algebra

I describe a method, particularly suitable to implementation by computer algebra, for the derivation of low-dimensional models of dynamical systems. The method is systematic and is based upon centre manifold theory. Computer code for the algorithm is relatively simple, robust and flexible. The method is applied to two examples: one a straightforward pitchfork bifurcation, and one being the dynamics of thin fluid films.

chao-dyn↗
arXiv1996

Unbiased estimation of multi-fractal dimensions of finite data sets

We present a novel method for determining multi-fractal properties from experimental data. It is based on maximising the likelihood that the given finite data set comes from a particular set of parameters in a multi-parameter family of well known multi-fractals. By comparing characteristic correlations obtained from the original data with those that occur in artificially generated multi-fractals with the {\em same} number of data points, we expect that predicted multi-fractal properties are unbiased by the finiteness of the experimental data.

chao-dyn↗
arXiv1994

The Swift-Hohenberg Equation Requires Non-Local Modifications to Model Spatial Pattern Evolution of Physical Problems

I argue that ``good'' mathematical models of spatio-temporal dynamics in two-dimensions require non-local operators in the nonlinear terms. Consequently, the often used Swift-Hohenberg equation requires modification as it is purely local. My aim here is to provoke more critical examination of the rationale for using the Swift-Hohenberg equations as a reliable model of the spatial pattern evolution in specific physical systems.

patt-sol↗
arXiv1994

The Construction of Zonal Models of Dispersion in Channels via Matched Centre Manifolds

Taylor's model of dispersion simply describes the long-term spread of material along a pipe, channel or river. However, often we need multi-mode models to resolve finer details in space and time. Here we construct zonal models of dispersion via the new principle of matching their long-term evolution with that of the original problem. Using centre manifold techniques this is done straightforwardly and systematically. Furthermore, this approach provides correct initial and boundary conditions for the zonal models. We expect the proposed principle of matched centre manifold evolution to be useful in a wide range of modelling problems.

chao-dyn↗