A quasinonlocal coupling method for nonlocal and local diffusion models

In this paper, we extend the idea of "geometric reconstruction" to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new coupling framework removes interfacial inconsistency, ensures the flux balance, and satisfies energy conservation as well as the maximum principle, whereas none of existing coupling methods for nonlocal-to-local coupling satisfies all of these properties. We establish the well-posedness and provide the stability analysis of the coupling method. We investigate the difference to the local limiting problem in terms of the nonlocal interaction range. Furthermore, we propose a first order finite difference numerical discretization and perform several numerical tests to confirm the theoretical findings. In particular, we show that the resulting numerical result is free of artifacts near the boundary of the domain where a classical local boundary condition is used, together with a coupled fully nonlocal model in the interior of the domain

Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media

We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.Comment: 30 pages, 21 figure

Rossby and Drift Wave Turbulence and Zonal Flows: the Charney-Hasegawa-Mima model and its extensions

A detailed study of the Charney-Hasegawa-Mima model and its extensions is presented. These simple nonlinear partial differential equations suggested for both Rossby waves in the atmosphere and also drift waves in a magnetically-confined plasma exhibit some remarkable and nontrivial properties, which in their qualitative form survive in more realistic and complicated models, and as such form a conceptual basis for understanding the turbulence and zonal flow dynamics in real plasma and geophysical systems. Two idealised scenarios of generation of zonal flows by small-scale turbulence are explored: a modulational instability and turbulent cascades. A detailed study of the generation of zonal flows by the modulational instability reveals that the dynamics of this zonal flow generation mechanism differ widely depending on the initial degree of nonlinearity. A numerical proof is provided for the extra invariant in Rossby and drift wave turbulence -zonostrophy and the invariant cascades are shown to be characterised by the zonostrophy pushing the energy to the zonal scales. A small scale instability forcing applied to the model demonstrates the well-known drift wave - zonal flow feedback loop in which the turbulence which initially leads to the zonal flow creation, is completely suppressed and the zonal flows saturate. The turbulence spectrum is shown to diffuse in a manner which has been mathematically predicted. The insights gained from this simple model could provide a basis for equivalent studies in more sophisticated plasma and geophysical fluid dynamics models in an effort to fully understand the zonal flow generation, the turbulent transport suppression and the zonal flow saturation processes in both the plasma and geophysical contexts as well as other wave and turbulence systems where order evolves from chaos.Comment: 64 pages, 33 figure

Quantum Plasmonics

Quantum plasmonics is an exciting subbranch of nanoplasmonics where the laws of quantum theory are used to describe light–matter interactions on the nanoscale. Plasmonic materials allow extreme subdiffraction confinement of (quantum or classical) light to regions so small that the quantization of both light and matter may be necessary for an accurate description. State-of-the-art experiments now allow us to probe these regimes and push existing theories to the limits which opens up the possibilities of exploring the nature of many-body collective oscillations as well as developing new plasmonic devices, which use the particle quality of light and the wave quality of matter, and have a wealth of potential applications in sensing, lasing, and quantum computing. This merging of fundamental condensed matter theory with application-rich electromagnetism (and a splash of quantum optics thrown in) gives rise to a fascinating area of modern physics that is still very much in its infancy. In this review, we discuss and compare the key models and experiments used to explore how the quantum nature of electrons impacts plasmonics in the context of quantum size corrections of localized plasmons and quantum tunneling between nanoparticle dimers. We also look at some of the remarkable experiments that are revealing the quantum nature of surface plasmon polaritons

Theory of Optical Nonlocality in Polar Dielectrics

Sub-wavelength confinement of mid-infrared light can be achieved exploiting the metal-like optical response of polar dielectric crystals in their Reststrahlen spectral region, where they support evanescent modes termed surface phonon polaritons. In the past few years the investigation of phonon polaritons localised in nanoresonators and layered heterostructures has enjoyed remarkable success, highlighting them as a promising platform for mid-infrared nanophotonic applications. Here we prove that the standard local dielectric description of phonon polaritons in nanometric objects fails due to the nonlocal nature of the phonon response and we develop the corresponding nonlocal theory. Application of our general theory to both dielectric nanospheres and thin films demonstrates that polar dielectrics exhibit a rich nonlocal phenomenology, qualitatively different from the one of plasmonic systems, due to the negative dispersion of phononic optical modes.Comment: 13 pages, 6 figure

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamics model away from the fracture set. The nonlocal model treated here is characterized by a double well potential and is a smooth version of the peridynamic model introduced in n Silling (J Mech Phys Solids 48(1), 2000). The solutions of nonlinear peridynamics are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale $\epsilon$ of non local interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation the consistency error for the numerical approximation is found to depend on the ratio between mesh size $h$ and $\epsilon$. More generally for local Lagrange interpolation of order $p\geq 1$ the consistency error is of order $h^p/\epsilon$. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size $h$ is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics