arXiv subjects

A. J. Roberts

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

At least 55 records · Page 3Linked to original sources

arXiv2012

Computer algebra compares the stochastic superslow manifold of an averaged SPDE with that of the original slow-fast SPDE

The computer algebra routines documented here empower you to reproduce and check many of the details described by an article on large deviations for slow-fast stochastic systems [abs:1001.4826]. We consider a 'small' spatial domain with two coupled concentration fields, one governed by a 'slow' reaction-diffusion equation and one governed by a stochastic 'fast' linear equation. In the regime of a stochastic bifurcation, we derive two superslow models of the dynamics: the first is of the averaged model of the slow dynamics derived via large deviation principles; and the second is of the original fast-slow dynamics. Comparing the two superslow models validates the averaging in the large deviation principle in this parameter regime.

math.DS↗

arXiv2012

Accurate macroscale modelling of spatial dynamics in multiple dimensions

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.

math.NA↗

arXiv2011

Self similarity and attraction in stochastic nonlinear reaction-diffusion systems

Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers' equation driven by some stochastic forced is studied. Under suitable assumptions, a stationary solution is constructed which yields the existence of a stochastic self-similar solution for the stochastic Burgers' equation. Furthermore, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems stochastic centre manifold theory provides a framework to construct the similarity solution, confirm its relevance, and determines the correct solution for any compact initial condition. Third, we argue that dynamically moving the spatial origin and dynamically stretching time improves the description of the stochastic similarity. Lastly, an application to an extremely simple model of turbulent mixing shows how anomalous fluctuations may arise in eddy diffusivities. The techniques and results we discuss should be applicable to a wide range of stochastic similarity problems.

math.DS↗

arXiv2011

On the approximation for singularly perturbed stochastic wave equations

We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations νu_{tt}+u_t=\D u+f(u)+ν^α\dot{W} on a bounded spatial domain. An asymptotic approximation to the stochastic wave equation is constructed by a special transformation and splitting of $νu_{t}$. This splitting gives a clear description of the structure of $u$. The approximating model, for small $ν>0$\,, is a stochastic nonlinear heat equation for exponent $0\leqα<1$\,, and is a deterministic nonlinear wave equation for exponent $α>1$\,.

math.AP↗

arXiv2011

Averaging approximation to singularly perturbed nonlinear stochastic wave equations

An averaging method is applied to derive effective approximation to the following singularly perturbed nonlinear stochastic damped wave equation νu_{tt}+u_t=\D u+f(u)+ν^α\dot{W} on an open bounded domain $D\subset\R^n$\,, $1\leq n\leq 3$\,. Here $ν>0$ is a small parameter characterising the singular perturbation, and $ν^α$\,, $0\leq α\leq 1/2$\,, parametrises the strength of the noise. Some scaling transformations and the martingale representation theorem yield the following effective approximation for small $ν$, u_t=\D u+f(u)+ν^α\dot{W} to an error of $\ord{ν^α}$\,.

math.AP↗

arXiv2011

Computer algebra derives the slow manifold of patch or element dynamics on lattices in two dimensions

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.

math.NA↗

arXiv2010

Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain

Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigourous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.

math.AP↗

arXiv2010

Resolve subgrid microscale interactions to discretise stochastic partial differential equations

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.

math.DS↗

arXiv2010

Large deviations for slow-fast stochastic partial differential equations

A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.

math.PR↗

arXiv2009

Average and deviation for slow-fast stochastic partial differential equations

Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of $\mathcal{O}(\e)$ instead of $\mathcal{O}(\sqrt{\e})$ attained in previous averaging.

math.AP↗

arXiv2009

Holistic discretisation ensures fidelity to dynamics in two spatial dimensions

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.

math.NA↗

arXiv2009

Slow manifold and averaging for slow-fast stochastic differential system

We consider multiscale stochastic dynamical systems. In this article an \emph{intermediate} reduced model is obtained for a slow-fast system with fast mode driven by white noise. First, the reduced stochastic system on exponentially attracting slow manifold reduced system is derived to errors of $\mathcal{O}(ε)$. Second, averaging derives an autonomous deterministic system up to errors of $\mathcal{O}(\sqrtε)$. Then an intermediate reduced model, which is an autonomous deterministic system driven by white noise up to errors of $\mathcal{O}(ε)$, is derived using a martingale approach to account for fluctuations about the averaged system. This intermediate reduced model has a simpler form than the reduced model on the stochastic slow manifold.

math-ph↗

arXiv2008

Macroscopic reduction for stochastic reaction-diffusion equations

The macroscopic behavior of dissipative stochastic partial differential equations usually can be described by a finite dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic reaction-diffusion equations with cubic nonlinearity by artificial separating the system into two distinct slow-fast time parts. An averaging method and a deviation estimate show that the macroscopic reduced model should be a stochastic ordinary equation which includes the random effect transmitted from the microscopic timescale due to the nonlinear interaction. Numerical simulations of an example stochastic heat equation confirms the predictions of this stochastic modelling theory. This theory empowers us to better model the long time dynamics of complex stochastic systems.

math-ph↗

arXiv2008

Choose interelement coupling to preserve self-adjoint dynamics in multiscale modelling and computation

Consider the macroscale modelling of microscale spatiotemporal dynamics. Here we develop a new approach to ensure coarse scale discrete models preserve important self-adjoint properties of the fine scale dynamics. The first part explores the discretisation of microscale continuum dynamics. The second addresses how dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids). Such mapping of discrete lattice dynamics may be iterated to empower us in future research to explore scale dependent emergent phenomena. The support of dynamical systems, centre manifold, theory ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain, and for all time. The accuracy of the models is limited by the asymptotic resolution of subgrid coarse scale processes, and is controlled by the level of truncation. As given examples demonstrate, the novel feature of the approach developed here is that it ensures the preservation of important conservation properties of the microscale dynamics.

nlin.CG↗

arXiv2008

Model turbulent floods with the Smagorinski large eddy closure

Floods, tides and tsunamis are turbulent, yet conventional models are based upon depth averaging inviscid irrotational flow equations. We propose to change the base of such modelling to the Smagorinksi large eddy closure for turbulence in order to appropriately match the underlying fluid dynamics. Our approach allows for large changes in fluid depth to cater for extreme inundations. The key to the analysis underlying the approach is to choose surface and bed boundary conditions that accommodate a constant turbulent shear as a nearly neutral mode. Analysis supported by slow manifold theory then constructs a model for the coupled dynamics of the fluid depth and the mean turbulent lateral velocity. The model resolves the internal turbulent shear in the flow and thus may be used in further work to rationally predict erosion and transport in turbulent floods.

nlin.CD↗

arXiv2008

Model dynamics on a multigrid across multiple length and time scales

Most methods for modelling dynamics posit just two time scales: a fast and a slow scale. But many applications, including many in continuum mechanics, possess a wide variety of space-time scales; often they possess a continuum of space-time scales. I discuss an approach to modelling the discretised dynamics of advection and diffusion with rigorous support for changing the resolved spatial grid scale by just a factor of two. The mapping of dynamics from a finer grid to a coarser grid is then iterated to generate a hierarchy of models across a wide range of space-time scales, all with rigorous support across the whole hierarchy. This approach empowers us with great flexibility in modelling complex dynamics over multiple scales.

nlin.CG↗

arXiv2007

The inertial dynamics of thin film flow of non-Newtonian fluids

Consider the flow of a thin layer of non-Newtonian fluid over a solid surface. I model the case of a viscosity that depends nonlinearly on the shear-rate; power law fluids are an important example, but the analysis here is for general nonlinear dependence. The modelling allows for large changes in film thickness provided the changes occur over a large enough lateral length scale. Modifying the surface boundary condition for tangential stress forms an accessible base for the analysis where flow with constant shear is a neutral critical mode, in addition to a mode representing conservation of fluid. Perturbatively removing the modification then constructs a model for the coupled dynamics of the fluid depth and the lateral momentum. For example, the results model the dynamics of gravity currents of non-Newtonian fluids even when the flow is not very slow.

math.DS↗

arXiv2007

Normal form transforms separate slow and fast modes in stochastic dynamical systems

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate all slow processes from all fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. The results will help us accurately model, interpret and simulate multiscale stochastic systems.

math.DS↗