Computational simulation is a key enabling technology in engineering, science and other quantitative fields. Coherent spatio-temporal dynamics, the main focus of application of this project, is the preeminent example of complex system behaviour as it emerges from the interactions of many similar components at each locale in space.
For the purposes of discussion consider that the microscopic model is one in the wide class of PDE's in the form
ut=Lu+f( u)+q(u,t)
(1)
where: x is position in one or more spatial dimensions; u(x,t) is some scalar or vector field, such as fluid velocity and pressure; L is a dissipative linear operator, such as Ñ2; f(u) includes other autonomous terms representing nonlinear advection, reaction, etc.; and q(u,t) is some time dependent control or possibly stochastic forcing [23]. Among many physically relevant examples are Burgers' equation [1,e.g.], the Brusselator [9,§3], Liouville's equation [13] and the Swift-Hohenberg equation [4,e.g.].
Consider forming a numerical solution of (1) by implementing the method of lines through discretising in the spatial variable x and integrating in time as a set of ordinary differential equations, sometimes called a semi-discrete scheme [5,6, e.g.]. A finite difference approximation to the spatial structure of (1) on a 1D regular grid is straightforward; for example, a linear diffusion term on a regular grid, say xj =jh for some grid spacing h, is
 2 u
x2
 =  uj+1-2uj+uj-1
h2
 +O(h2)  ,
where uj is the value of u at the grid points xj. However, there are many differing valid alternatives for a nonlinear term such as the self-advection uux: two possibilities are
u  u
x
 =  uj(uj+1-uj-1)
2h
 +O(h2) =  uj+12-uj-12
4h
 +O(h2)  ;
and a third is the 1:2 mix of the above two suggested by Fornberg [7] and used to improve stability [5,e.g.]. The best choice depends upon how the discretisation of the nonlinearity interacts with the dynamics of other terms. The traditional approach of considering the discretisation of each term in the equation separately does not answer. To find the best discretisation we consider the complex interaction of all terms in the PDE (1) in a "holistic" approach that involves resolving microscale subgrid structures and their evolution. Centre manifold techniques construct approximations based upon the principle of capturing an exponentially attractive manifold of actual solutions. The approximation is consequently invariant under any valid algebraic rewriting of the governing equations and preserves symmetries. Jones [11] argue that quite generally there exist good approximations to such attractive inertial manifolds. One core challenge of this project is to actually construct effective approximations and to give general theoretical support so that modern ideas become practical tools for engineers and scientists.
Centre manifold theory is a powerful tool for the modelling of complex dynamical systems [17,19,8,e.g.] such as dispersion [25,16,e.g.], thin fluid films [2,20,26,e.g.], stochastic systems [28,18], control [12,e.g.] and turbulent floods [15]. Based upon modifying linear dynamics the theory guarantees that an accurate and relevant low-dimensional description of the nonlinear dynamics may be deduced. We currently place the discretisation of a nonlinear PDE such as (1) within the purview of centre manifold theory by the following artifice; such adaptation also proves effective in thin fluid flows [20] and dispersion [24]. Tessellate the spatial domain into finite size elements, say there are m elements of size h, and then introduce a homotopy parameter g, 0 £ g £ 1, parametrising "internal boundary conditions" ( IBC) [13,21] between each element:
 uo
n
 =  ui
n
  ,    (1-g)  h
2
 æ
è 		 uo
n
 +  ui
n
 ö
ø 		=g( uo-ui) ,
(2)
and its higher order analogues [14], where [()/(n)] is the derivative normal to the boundary, ui is from the element under consideration (inside) and uo is from the adjacent element (outside). When g = 1 these reduce to conditions ensuring appropriate continuity between adjacent elements. When g = 0 they reduce to effectively insulating conditions. We then treat terms multiplied by g as "nonlinear" perturbations to the insulated dynamics. Thus in the "linear" dissipative dynamics governed by ut=Lu the field u in each element evolves exponentially quickly (typically in a time O(h2) for a diffusive system) to some constant value in the jth element, say u=uj . But in the presence of the nonlinear terms and the coupling between the elements when g ¹ 0, the values uj associated with each element evolve in time. Centre manifold theory assures three things for the system of coupled elements: the existence of an m dimensional slow manifold parametrised by grid values uj; the relevance of the m dimensional dynamics as an accurate and stable model of the original dynamics (1); and that we may systematically approximate the subgrid scale structures and the corresponding macroscale evolution. Importantly, symmetries of the PDE (1), compatible with the IBC (2), are fully maintained in the analysis. These dynamics on the slow manifold form a sound and systematic computational model on the macroscale h.
It might be argued that the dynamics of the PDE (1) recovered at g = 1 is unrelated to that of the derived discrete model which is based upon asymptotics about g = 0. But we routinely use asymptotic expansions at finite values of a notionally small parameter. The practical issue here is whether the expansion converges at g = 1 . In application, our approach converges to a global spectral discretisation of Burgers' equation [21,Appendix]. We additionally ensure high order consistency in the limit h® 0 [22] by the effectively near identity modification of the IBC's (2) to
uo(xj±1)-ui(xj)=g[ui(xj±1)-ui(xj)] .
(3)
That the modelling process satisfies these two independent asymptotic limits provides wonderful support for the proposed approach. For example, in Burgers' equation it seems best to discretise the nonlinear advection uux [21] with a higher order correction which automatically improves the stability and accuracy of the numerical model. Indeed following Foias [5] [§2.1] we found our holistic discretisation is nonlinearly stable when other discretisations are not. We need to be inventive in exploring and then analysing various options for the form of the coupling in order to ensure best performance of the resulting computational models.
This coupling between elements is a fundamental issue for multiscale modelling in general. Modification of the coupling of patches in the gap-tooth scheme [10,27] achieves higher order consistency by being asymptotically consistent with (3). Further, the extension of the subgrid fields outside of the element, ui(xj±1) in (3) is analogous to the solution outside of their finite elements which Chen [3] required in their multiscale modelling.

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Copyright Tony Roberts, 5 Nov 2006
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