Analysis of Iterative Methods for the Linear Boltzmann Transport Equation

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the discretisation parameters. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport.Comment: 27 pages, 8 figure

A 2D extension of a large time step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries

A 2D Large Time Step (LTS) explicit scheme on structured grids is presented in this work. It is first detailed and analysed for the 2D linear advection equation and then applied to the 2D shallow water equations. The dimensional splitting technique allows us to extend the ideas developed in the 1D case related to source terms, boundary conditions and the reduction of the time step in the presence of large discontinuities. The boundary conditions treatment as well as the wet/dry fronts in the case of the 2D shallow water equations require extra effort. The proposed scheme is tested on linear and non-linear equations and systems, with and without source terms. The numerical results are compared with those of the conventional scheme as well as with analytical solutions and experimental data

Worcester’s Missing Political Voice and the Fate of the Auditorium

The political activism of the 25,000 Worcester college students has been rare given the potential power of numbers. Even on issues that directly affect them they exert no influence. This paper reports a case study of our effort to recruit students to attend a city wide charrette with a focus on Lincoln Square. Prior research indicates that the college students of Worcester want a gathering place and one proposal for the Memorial Auditorium dominating Lincoln Square is to make it a College Crossroads . The East Highland Neighborhood Association supports this proposal for the Auditorium and sponsored our effort to determine if the students will mobilize for this cause. We also assessed support for several other uses of the Auditorium that the adjacent neighborhood might support

Effective equations governing an active poroelastic medium

In this work we consider the spatial homogenization of a coupled transport and fluid-structure interaction model, to the end of deriving a system of effective equations describing the flow, elastic deformation, and transport in an active poroelastic medium. The `active' nature of the material results from a morphoelastic response to a chemical stimulant, in which the growth timescale is strongly separated from other elastic timescales. The resulting effective model is broadly relevant to the study of biological tissue growth, geophysical flows (e.g. swelling in coals and clays) and a wide range of industrial applications (e.g. absorbant hygiene products). The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscale growth, coupled to transport of solute, that explicitly incorporates details of the structure and dynamics of the microscopic system, and, moreover, admits finite growth and deformation at the pore-scale. The resulting macroscale model comprises a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection-reaction-diffusion equation. The resultant system of effective equations is then compared to other recent models under a selection of appropriate simplifying asymptotic limits

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an $hp$-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and $hp$-version a priori error analysis of the proposed method, by deriving suitable $hp$-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.Comment: 27 pages, 2 figure

Dark-field transmission electron microscopy and the Debye-Waller factor of graphene

Graphene's structure bears on both the material's electronic properties and fundamental questions about long range order in two-dimensional crystals. We present an analytic calculation of selected area electron diffraction from multi-layer graphene and compare it with data from samples prepared by chemical vapor deposition and mechanical exfoliation. A single layer scatters only 0.5% of the incident electrons, so this kinematical calculation can be considered reliable for five or fewer layers. Dark-field transmission electron micrographs of multi-layer graphene illustrate how knowledge of the diffraction peak intensities can be applied for rapid mapping of thickness, stacking, and grain boundaries. The diffraction peak intensities also depend on the mean-square displacement of atoms from their ideal lattice locations, which is parameterized by a Debye-Waller factor. We measure the Debye-Waller factor of a suspended monolayer of exfoliated graphene and find a result consistent with an estimate based on the Debye model. For laboratory-scale graphene samples, finite size effects are sufficient to stabilize the graphene lattice against melting, indicating that ripples in the third dimension are not necessary.Comment: 10 pages, 4 figure

Mathematical and computational models of drug transport in tumours

The ability to predict how far a drug will penetrate into the tumour microenvironment within its pharmacokinetic (PK) lifespan would provide valuable information about therapeutic response. As the PK profile is directly related to the route and schedule of drug administration, an in silico tool that can predict the drug administration schedule that results in optimal drug delivery to tumours would streamline clinical trial design. This paper investigates the application of mathematical and computational modelling techniques to help improve our understanding of the fundamental mechanisms underlying drug delivery, and compares the performance of a simple model with more complex approaches. Three models of drug transport are developed, all based on the same drug binding model and parametrized by bespoke in vitro experiments. Their predictions, compared for a ‘tumour cord’ geometry, are qualitatively and quantitatively similar. We assess the effect of varying the PK profile of the supplied drug, and the binding affinity of the drug to tumour cells, on the concentration of drug reaching cells and the accumulated exposure of cells to drug at arbitrary distances from a supplying blood vessel. This is a contribution towards developing a useful drug transport modelling tool for informing strategies for the treatment of tumour cells which are ‘pharmacokinetically resistant’ to chemotherapeutic strategies

The Aboveground and Belowground Growth Characteristics of Juvenile Conifers in the Southwestern United States

Juvenile tree survival will play an important role in the persistence of coniferous forests and woodlands in the southwestern United States (SWUS). Vulnerability to climatic and environmental stress declines as trees grow, such that larger, more deeply rooted juveniles are less likely to experience mortality. It is unclear how juvenile conifers partition the aboveground and belowground components of early growth, if growth differs between species and ecosystem types, and what environmental factors influence juvenile carbon allocation above- or belowground. We developed a novel data set for four juvenile conifer groups (junipers, piñon pines, ponderosa pines, firs; 1121 juveniles sampled, 221 destructively) in three height classes ( \u3c 150 mm, 150–300 mm, and 300+ mm), across 25 SWUS sites. We compared growth characteristics across groups and height classes and related differences to climatic and environmental factors. As tree height increased from \u3c 150 mm to 300+ mm, belowground growth increased, root:shoot ratio declined, and specific leaf area declined for all conifers except firs. Maximum rooting depth was shallower than previous estimates ( \u3c ˜400 mm). Lower elevation juveniles were frequently located in sheltered microsites that provided high shading, whereas mid- and higher elevation juveniles were frequently unsheltered. Across all forest and woodland sites, herbaceous cover was positively correlated with aboveground growth. At study locations comprised of multiple sites, differences in aboveground growth were best explained by ecosystem type (piñon pine-juniper woodland, ponderosa pine forest, mixed-conifer forest) and local environmental variation. Our results indicate generally more belowground early growth and more aboveground later growth, but specific allocation patterns varied among ecosystem (greater proportional shoot growth at lower and mid-elevations compared with higher elevations). Juvenile conifers had similar magnitudes of proportional growth across conifer groups, displaying limited capacity to acclimate growth to differences in climate that control ecosystem type. If juvenile conifers also do not acclimate physiologically to their environment, our findings suggest that local environmental variation will play a primary role in regulating forest and woodland persistence and modify the effects of climate change in the SWUS

Dark-field transmission electron microscopy and the Debye-Waller factor of graphene

Graphene\u27s structure bears on both the material\u27s electronic properties and fundamental questions about long-range order in two-dimensional crystals. We present an analytic calculation of selected area electron diffraction from multilayer graphene and compare it with data from samples prepared by chemical vapor deposition and mechanical exfoliation. A single layer scatters only 0.5% of the incident electrons, so this kinematical calculation can be considered reliable for five or fewer layers. Dark-field transmission electron micrographs of multilayer graphene illustrate how knowledge of the diffraction peak intensities can be applied for rapid mapping of thickness, stacking, and grain boundaries. The diffraction peak intensities also depend on the mean-square displacement of atoms from their ideal lattice locations, which is parameterized by a Debye-Waller factor. We measure the Debye-Waller factor of a suspended monolayer of exfoliated graphene and find a result consistent with an estimate based on the Debye model. For laboratory-scale graphene samples, finite size effects are sufficient to stabilize the graphene lattice against melting, indicating that ripples in the third dimension are not necessary