Holistic discretisation of shear dispersion in a two-dimensional channel

MacKenzie, T.; Roberts, A. J.

doi:10.21914/anziamj.v44i0.694

Cited by 10 publications

(19 citation statements)

References 12 publications

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“…The same methodology, but with different details will account for physical boundaries to produce a discrete model valid across the whole domain. Such modelling incorporating physical boundaries has already been shown for the deterministic Burgers' pde [44] and shear dispersion in a channel [45].…”

Section: Resultsmentioning

confidence: 82%

Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations

Roberts¹

2006

LMS J. Comput. Math.

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Constructing discrete models of stochastic partial differential equations is very delicate. Stochastic centre manifold theory provides novel support for coarse grained, macroscale, spatial discretisations of nonlinear stochastic partial differential or difference equations such as the example of the stochastically forced Burgers' equation. Dividing the physical domain into finite length overlapping elements empowers the approach to resolve fully coupled dynamical interactions between neighbouring elements. The crucial aspect of this approach is that the underlying theory organises the resolution of the vast multitude of subgrid microscale noise processes interacting via the nonlinear dynamics within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in the spatial discretisation. Nonlinear interactions have two further consequences: additive forcing generates multiplicative noise in the discretisation; and effectively new noise processes appear in the macroscale discretisation. The techniques and theory developed here may be applied to soundly discretise many dissipative stochastic partial differential and difference equations.

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Section: Resultsmentioning

confidence: 82%

Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations

Roberts¹

2006

LMS J. Comput. Math.

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“…This approach to spatial discretisation of the npde (1) may be extended to higher spatial dimensions as for autonomous pdes [26,44]. Because of the need to decompose the residuals into eigenmodes on each element, the application to higher spatial dimensions are likely to require tessellating space into simple regular elements for npdes.…”

Section: Resultsmentioning

confidence: 99%

Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations

Bunder

Roberts

2017

Applied Mathematics and Computation

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Coarse grained, macroscale, spatial discretisations of nonlinear nonautonomous partial differential/difference equations are given novel support by centre manifold theory. Dividing the physical domain into overlapping macroscale elements empowers the approach to resolve significant subgrid microscale structures and interactions between neighbouring elements. The crucial aspect of this approach is that centre manifold theory organises the resolution of the detailed subgrid microscale structure interacting via the nonlinear dynamics within and between neighbouring elements. The techniques and theory developed here may be applied to soundly discretise on a macroscale many dissipative nonautonomous partial differential/difference equations, such as the forced Burgers' equation, adopted here as an illustrative example.This article develops a systematic approach to constructing spatially discrete macroscale models of nonlinear nonautonomous partial differential/difference equations (collectively denoted nonlinear npdes). In particular, we address issues arising from rich and significant microstructure on a considerably smaller spatial scale than the macroscale grid. Although this article concentrates on

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“…For example, at the beginning of the third line in (13) see the term −cH(γ − γ 2 ) 1 2 ξδ 2 disappears when we set γ = 1 for the physically relevant approximation. Similarly, computing the next order terms in coupling parameter γ generates terms, in γ 3 , which cancel the r dependent terms in the third and fourth line of the subgrid field (13). Thus higher order models push any undesirable r dependence to higher orders, thereby usefully predicting a subgrid field largely independent of the patch size r.…”

Section: C648mentioning

confidence: 99%

“…The method of "holistic discretisation", developed by Roberts & Mackenzie [17,18,19,13], creates discretisations on a macroscopic grid using systematically obtained analytic approximations for the subgrid field. The analytic solutions of Section 3 using this method are analogous to the microscopic system simulators in the gap-tooth scheme: they both provide microscopic solutions which are macroscopically coupled to neighbouring elements.…”

Section: C641mentioning

confidence: 99%

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

Roberts¹,

Kevrekidis²

2005

ANZIAMJ

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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 begun here is to find general boundary conditions for the patches of microscopic simulators that appropriately connect the widely separated "teeth" to * Dept. ANZIAM J. 46 (E) pp.C637-C657, 2005C638 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. In 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.

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Holistic discretisation of shear dispersion in a two-dimensional channel

Cited by 10 publications

References 12 publications

Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations

Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations

Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations

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

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