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.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv admin note: substantial text overlap with arXiv:0904.085

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

A multiscale hybrid approach for vasculogenesis and related potential blocking therapies

Solid tumors must recruit and form new blood vessels for maintenance, growth and detachments of metastases. Discovering drugs that block malignant angiogenesis is thus an important approach in cancer treatment and has given rise to multiple in vitro and in silico models. The present hybrid individual cell-based model incorporates some underlying biochemical events relating more closely the classical Cellular Potts Model (CPM) parameters to subcellular mechanisms and to the activation of specific signaling pathways. The model spans the three fundamental biological levels: at the extracellular level a continuous model describes secretion, diffusion, uptake and decay of the autocrine VEGF; at the cellular level, an extended lattice CPM, based on a system energy reduction, reproduces cell dynamics such as migration, adhesion and chemotaxis; at the subcellular level, a set of reaction-diffusion equations describes a simplified VEGF-induced calcium-dependent intracellular pathway. The results agree with the known interplay between calcium signals and VEGF dynamics and with their role in malignant vasculogenesis. Moreover, the analysis of the link between the microscopic subcellular dynamics and the macroscopic cell behaviors confirms the efficiency of some pharmacological interventions that are currently in use and, more interestingly, proposes some new therapeutic approaches, that are counter-intuitive but potentially effective

The role of disorder in the electronic and transport properties of monolayer and bilayer graphene

This thesis is devoted to the computational study of the electronic and transport properties of monolayer and bilayer graphene in the presence of disorder arising from both topological and point defects. Among the former, we study grain boundaries in monolayer graphene and stacking domain boundaries in bilayer graphene, whereas among the latter we study hydrogen atoms covalently bound on the graphene crystal lattice. The electronic spectrum of disordered graphene has been studied within a tight-binding framework, which has been coupled to the Landauer-Büttiker theory and Green¿s function techniques in order to have access to the properties of coherent transport of graphene charge carriers. We assess the low-energy equilibrium structures of defective graphene by a combination of ab initio density functional theory, classical potentials, and Monte Carlo methods. We study periodic grain boundaries in monolayer graphene and individuate two classes of defects with opposite effects in terms of scattering of low-energy charge carriers. One class, unexpectedly, is highly reflecting in the limit of low defect density, whereas another is highly transparent. Subsequently, we study disordered grain boundaries in order to predict the intrinsic conductance of realistic polycrystalline graphene samples. In two related works, conducted in collaboration with experimentalists, we identify the atomic structure of periodic grain boundaries imaged by scanning tunneling microscopy, and discuss the valley-filtering capabilities of a line defect of graphene that can be grown in a controllable manner. Next, we investigate the electronic transport of graphene with realistic hydrogen adsorbates, whose equilibrium configurations are obtained by means of Monte Carlo simulations. We find that the conductance of graphene dramatically increases upon formation of cluster adatoms, which we predict to happen spontaneously at room temperature. This is due to the non- resonant nature of a large fraction of hydrogen clusters in the room-temperature distribution, which we further elucidate by means of an analytically solvable model. Finally, we study the behavior, in terms of structural and electronic properties, of twisted bilayer graphene in the limit of zero twist angle. We find a critical angle below which the system arranges in a triangular superlattice of Bernal-stacking domains, separated by a hexagonal network of stacking domain boundaries. The presence of stacking domain boundaries is at the base of our interpretation of an experiment reporting oscillations in the electrical conductance of bilayer graphene subjected to mechanical indentation