Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale system at a desired macroscale system level. In this scheme, the given microscale model is computed in s
mall patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate simulations, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve translational invariance, rotational invariance, and self-adjoint symmetry, thus guaranteeing that conservation laws associated with these symmetries are preserved in the macroscale simulation. Spectral and algebraic analyses of the proposed scheme in both one and two dimensions reveal mechanisms for further improving the accuracy of the simulations. Consistency of the patch schemes macroscale dynamics with the original microscale model is proved. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation in a wide range of multiscale scenarios of interest to scientists and engineers.
Scientists and engineers often create accurate, trustworthy, computational simulation schemes - but all too often these are too computationally expensive to execute over the time or spatial domain of interest. The equation-free approach is to marry s
uch trusted simulations to a framework for numerical macroscale reduction - the patch dynamics scheme. This article extends the patch scheme to scenarios in which the trusted simulation resolves abrupt state changes on the microscale that appear as shocks on the macroscale. Accurate simulation for problems in these scenarios requires extending the patch scheme by capturing the shock within a novel patch, and also modifying the patch coupling rules in the vicinity in order to maintain accuracy. With these two extensions to the patch scheme, straightforward arguments derive consistency conditions that match the usual order of accuracy for patch schemes. The new scheme is successfully tested on four archetypal problems. This technique will empower scientists and engineers to accurately and efficiently simulate, over large spatial domains, multiscale multiphysics systems that have rapid transition layers on the microscale.
The `equation-free toolbox empowers the computer-assisted analysis of complex, multiscale systems. Its aim is to enable you to immediately use microscopic simulators to perform macro-scale system level tasks and analysis, because micro-scale simulati
ons are often the best available description of a system. The methodology bypasses the derivation of macroscopic evolution equations by computing the micro-scale simulator only over short bursts in time on small patches in space, with bursts and patches well-separated in time and space respectively. We introduce the suite of coded equation-free functions in an accessible way, link to more detailed descriptions, discuss their mathematical support, and introduce a novel and efficient algorithm for Projective Integration. Some facets of toolbox development of equation-free functions are then detailed. Download the toolbox functions (https://github.com/uoa1184615/EquationFreeGit) and use to empower efficient and accurate simulation in a wide range of your science and engineering problems.
Many physical systems are formulated on domains which are relatively large in some directions but relatively thin in other directions. We expect such systems to have emergent structures that vary slowly over the large dimensions. Common mathematical
approximations for determining the emergent dynamics often rely on self-consistency arguments or limits as the aspect ratio of the `large and `thin dimensions becomes nonphysically infinite. Here we extend to nonlinear dynamics a new approach [IMA J. Appl. Maths, DOI: 10.1093/imamat/hxx021] which analyses the dynamics at each cross-section of the domain via a rigorous multivariate Taylor series. Then centre manifold theory supports the global modelling of the systems emergent dynamics in the large but finite domain. Interactions between the cross-section coupling and both fast and slow dynamics determines quantitative error bounds for the nonlinear modelling. We illustrate the methodology by deriving the large-scale dynamics of a thin liquid film, where the film is subject to a Coriolis force induced by a rotating substrate. The approach developed here quantifies the accuracy of known approximations, extends such approximations to mixed order modelling, and may open previously intractable modelling issues to new tools and insights.
This article establishes the foundation for a new theory of invariant/integral manifolds for non-autonomous dynamical systems. Current rigorous support for dimensional reduction modelling of slow-fast systems is limited by the rare events in stochast
ic systems that may cause escape, and limited in many applications by the unbounded nature of PDE operators. To circumvent such limitations, we initiate developing a backward theory of invariant/integral manifolds that complements extant forward theory. Here, for deterministic non-autonomous ODE systems, we construct a conjugacy with a normal form system to establish the existence, emergence and exact construction of center manifolds in a finite domain for systems `arbitrarily close to that specified. A benefit is that the constructed invariant manifolds are known to be exact for systems `close to the one specified, and hence the only error is in determining how close over the domain of interest for any specific application. Built on the base developed here, planned future research should develop a theory for stochastic and/or PDE systems that is useful in a wide range of modelling applications.
Many physical systems are well described on domains which are relatively large in some directions but relatively thin in other directions. In this scenario we typically expect the system to have emergent structures that vary slowly over the large dim
ensions. For practical mathematical modelling of such systems we require efficient and accurate methodologies for reducing the dimension of the original system and extracting the emergent dynamics. Common mathematical approximations for determining the emergent dynamics often rely on self-consistency arguments or limits as the aspect ratio of the large and thin dimensions becomes unphysically infinite. Here we build on a new approach, previously establish for systems which are large in only one dimension, which analyses the dynamics at each cross-section of the domain with a rigorous multivariate Taylor series. Then centre manifold theory supports the local modelling of the systems emergent dynamics with coupling to neighbouring cross-sections treated as a non-autonomous forcing. The union over all cross-sections then provides powerful support for the existence and emergence of a centre manifold model global in the large finite domain. Quantitative error estimates are determined from the interactions between the cross-section coupling and both fast and slow dynamics. Two examples provide practical details of our methodology. The approach developed here may be used to quantify the accuracy of known approximations, to extend such approximations to mixed order modelling, and to open previously intractable modelling issues to new tools and insights.
Multiscale modelling aims to systematically construct macroscale models of materials with fine microscale structure. However, macroscale boundary conditions are typically not systematically derived, but rely on heuristic arguments, potentially result
ing in a macroscale model which fails to adequately capture the behaviour of the microscale system. We derive the macroscale boundary conditions of the macroscale model for longitudinal wave propagation on a lattice with periodically varying density and elasticity. We model the macroscale dynamics of the microscale Dirichlet, Robin-like, Cauchy-like and mixed boundary value problem. Numerical experiments test the new methodology. Our method of deriving boundary conditions significantly improves the accuracy of the macroscale models. The methodology developed here can be adapted to a wide range of multiscale wave propagation problems.
Finite difference/element/volume methods of discretising PDEs impose a subgrid scale interpolation on the dynamics. In contrast, the holistic discretisation approach developed herein constructs a natural subgrid scale field adapted to the whole syste
m out-of-equilibrium dynamics. Consequently, the macroscale discretisation is fully informed by the underlying microscale dynamics. We establish a new proof that in principle there exists an exact closure of the dynamics of a general class of reaction-advection-diffusion PDEs, and show how our approach constructs new systematic approximations to the in-principle closure starting from a simple, piecewise-linear, continuous approximation. Under inter-element coupling conditions that guarantee continuity of several field properties, the holistic discretisation possesses desirable properties such as a natural cubic spline first-order approximation to the field, and the self-adjointness of the diffusion operator under periodic, Dirichlet and Neumann macroscale boundary conditions. As a concrete example, we demonstrate the holistic discretisation procedure on the well-known Burgers PDE, and compare the theoretical and numerical stability of the resulting discretisation to other approximations. The approach developed here promises to be able to systematically construct automatically good, macroscale discretisations to a wide range of PDEs, including wave PDEs.
Multiscale modelling methodologies build macroscale models of materials with complicated fine microscale structure. We propose a methodology to derive boundary conditions for the macroscale model of a prototypical non-linear heat exchanger. The deriv
ed macroscale boundary conditions improve the accuracy of macroscale model. We verify the new boundary conditions by numerical methods. The techniques developed here can be adapted to a wide range of multiscale reaction-diffusion-advection systems.
Massive parallelisation has lead to a dramatic increase in available computational power. However, data transfer speeds have failed to keep pace and are the major limiting factor in the development of exascale computing. New algorithms must be develo
ped which minimise the transfer of data. Patch dynamics is a computational macroscale modelling scheme which provides a coarse macroscale solution of a problem defined on a fine microscale by dividing the domain into many nonoverlapping, coupled patches. Patch dynamics is readily adaptable to massive parallelisation as each processor can evaluate the dynamics on one, or a few, patches. However, patch coupling conditions interpolate across the unevaluated parts of the domain between patches, and are typically reevaluated at every microscale time step, thus requiring almost continuous data transfer. We propose a modified patch dynamics scheme which minimises data transfer by only reevaluating the patch coupling conditions at `mesoscale time scales which are significantly larger than the microscale time of the microscale problem. We analyse the error arising from patch dynamics with mesoscale temporal coupling as a function of the mesoscale time interval, patch size, and ratio between the microscale and macroscale.
