A. J. Roberts

A. J. Roberts

A practical and new Runge--Kutta numerical scheme for stochastic differential equations is explored. Numerical examples demonstrate the strong convergence of the method. The first order strong convergence is then proved using Ito integrals for both Ito and Stratonovich interpretations. As a straightforward modification of the deterministic Improved Euler/Heun method, the method is a good entry level scheme for stochastic differential equations, especially in conjunction with Higham's introduction [SIAM Review, 43:525--546, 2001].

A. J. Roberts, Tony MacKenzie, J. E. Bunder

Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid microscale dynamics the dynamical systems approach constructs accurate closures of macroscale discretisations of the microscale system. Here we specifically explore reaction-diffusion problems in two spatial dimensions as a prototype of generic systems in multiple dimensions. Our approach unifies into one the modelling of systems by a type of finite elements, and the `equation free' macroscale modelling of microscale simulators efficiently executing only on small patches of the spatial domain. Centre manifold theory ensures that a closed model exist on the macroscale grid, is emergent, and is systematically approximated. Dividing space either into overlapping finite elements or into spatially separated small patches, the specially crafted inter-element/patch coupling also ensures that the constructed discretisations are consistent with the microscale system/PDE to as high an order as desired. Computer algebra handles the considerable algebraic details as seen in the specific application to the Ginzburg--Landau PDE. However, higher order models in multiple dimensions require a mixed numerical and algebraic approach that is also developed. The modelling here may be straightforwardly adapted to a wide class of reaction-diffusion PDEs and lattice equations in multiple space dimensions. When applied to patches of microscopic simulations our coupling conditions promise efficient macroscale simulation.

G. A. Jarrad, A. J. Roberts

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 system 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.

Tony MacKenzie, A. J. Roberts

Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. Here we explore the methodology applied to the gap-tooth scheme in the equation-free approach of Kevrekidis in two spatial dimensions. The algebraic detail is enormous so we detail computer algebra procedures to handle the enormity. However, modelling the dynamics on 2D spatial patches appears to require a mixed numerical and algebraic approach that is detailed in this report. Being based upon the computation of residuals, the procedures here may be simply adapted to a wide class of reaction-diffusion equations.

J. E. Bunder, A. J. Roberts, I. G. Kevrekidis

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 developed 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.

A. J. Roberts

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.

Tony MacKenzie, A. J. Roberts

Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate, macroscale closures of discrete models of the \pde. Here we explore reaction-diffusion problems in two spatial dimensions. Centre manifold theory ensures that slow manifold, holistic, discretisations exists, are quickly attractive, and are systematically approximated. Special coupling of the finite elements ensures that the resultant discretisations are consistent with the \pde to as high an order as desired. Computer algebra handles the enormous algebraic details as seen in the specific application to the Ginzburg--Landau equation. However, higher order models in 2D appear to require a mixed numerical and algebraic approach that is also developed. Being driven by the residuals of the equations, the modelling here may be straightforwardly adapted to a wide class of reaction-diffusion differential and lattice equations in multiple space dimensions.

A. J. Roberts, I. G. Kevrekidis

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.

T. MacKenzie, A. J. Roberts

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.

A. J. Roberts, I. G. Kevrekidis

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.