A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain

Fast computation of effective diffusivities using a semi-analytical solution of the homogenization boundary value problem for block locally-isotropic heterogeneous media

Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution method to a standard finite volume method and show that equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new solution method can be applied to complex heterogeneous geometries represented by a grid of blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up simulations of diffusion in heterogeneous media.Comment: 29 pages, 4 figures, 5 table

Foam patches behind spilling breakers

Previous theoretical and laboratory studies of spilling breakers on a beach are described and discussed, paying particular attention to models that emphasize the importance of air bubbles in the spill. At first such a spill forms at a sharp crest, and accelerates down the front of the wave as it propagates towards the shore. Then the crest becomes more rounded and this allows part of the aerated water to flow backwards over or under the crest, leaving a foam patch behind. Laboratory experiments in channels of constant width have documented many features of these flows. But recent new observations on gently sloping beaches have revealed another three-dimensional phenomenon, which is not possible in laboratory channels, and which has rarely been described or properly explained. Initially the spill forms nearly simultaneously over a wide front and extends down the forward face of the wave. Then at some point on the spill some of the foam flows over the crest and is left behind in a continuous patch of increasing length. At the same time the rounded portion of the crest propagates sideways in both directions, as the shear flow at its edge entrains fluid from the sharp crests on either side. This leads to a patch that is triangular in shape, with a peak directly behind the initial instability and of zero length where the backward flow has just begun. This idea has been quantitatively tested using selected photos taken from a headland above a beach, rectified to produce plan views. The patches are indeed triangular, sometimes distorted by a shear flow parallel to the wave crest but with a narrow range of peak angles, and on this beach which has a very uniform slope there is no systematic dependence on other parameters such as the wave height at breaking or the bathymetry conditions

Lonely heart columns: A novel and entertaining way of teaching students abstract writing skills

Abstract writing is a key skill for science graduates; they are a common feature in many of the standard forms of scientific dissemination such as scientific research articles. In this paper we present a novel and entertaining approach for teaching abstract writing using adverts from lonely heart columns (LHC). Student constructed full profiles of the authors of LHC and constructed LHC profiles of celebrities to illustrate the key sills in abstract construction. There was no significant difference between the grades achieved by student taught using LHC and a more traditional approach, suggesting there were no negative impacts from this delivery method. Student in LHC tutorial overwhelmingly enjoy the tutorial, 95% responded the question ‘how would you rate the enjoyment of this tutorial’ as ‘much’ or ‘very much’. In addition to abstract writing two thirds of students in LHC tutorial believed they improved their ability to speak in front of others and their creative thinking skills. The LHC tutorial is a novel approach to teaching and learning that is both enjoyable and effective

Numerical investigation into coarse-scale models of diffusion in complex heterogeneous media

Computational modelling of diffusion in heterogeneous media is prohibitively expensive for problems with fine-scale heterogeneities. A common strategy for resolving this issue is to decompose the domain into a number of non-overlapping sub-domains and homogenize the spatially-dependent diffusivity within each sub-domain (homogenization cell). This process yields a coarse-scale model for approximating the solution behaviour of the original fine-scale model at a reduced computational cost. In this paper, we study coarse-scale diffusion models in block heterogeneous media and investigate, for the first time, the effect that various factors have on the accuracy of resulting coarse-scale solutions. We present new findings on the error associated with homogenization as well as recommendations for choosing the number of homogenization cells and the type of boundary condition imposed on the homogenization cells to produce accurate coarse-scale solutions.Comment: 19 pages, 10 figures, submitte

Fe XXV and Fe XXVI Diagnostics of the Black Hole and Accretion Disk in Active Galaxies: Chandra Time-Resolved Spectroscopy of NGC 7314

We report the detection of Fe XXV and Fe XXVI $K\alpha$ emission lines from a Chandra HETGS observation of the Seyfert~1 galaxy NGC 7314, made simultaneously with RXTE. The lines are redshifted (cz ~ 1500 km/s) relative to the systemic velocity and unresolved. We argue that the lines originate in a near face-on (<7 degrees) disk having a radial line emissivity flatter than r^-2. Line emission from ionization states of Fe in the range ~Fe I up to Fe XXVI is observed. The ionization balance of Fe responds to continuum variations on timescales less than 12.5 ks, supporting an origin of the lines close to the X-ray source. We present additional, detailed diagnostics from this rich data set. These results identify NGC 7314 as a key source to study in the future if we are to pursue reverberation mapping of space-time near black-hole event horizons. This is because it is first necessary to understand the ionization structure of accretion disks and the relation between the X-ray continuum and Fe K line emission. However, we also describe how our results are suggestive of a means of measuring black-hole spin without a knowledge of the relation between the continuum and line emission. Finally, these data emphasize that one {\it can} study strong gravity with narrow (as opposed to very broad) disk lines. In fact narrow lines offer higher precision, given sufficient energy resolution.Comment: Accepted for publication in the Astrophysical Journal. 30 pages, six figures, five of them color. Abstract is abridge

Dynamic restoration and the impact of native versus invasive vegetation on coastal foredune morphodynamics, Lanphere Dunes, California, USA

The Lanphere Dunes, part of the Humboldt Bay National Wildlife Refuge, has been the focus of foredune restoration efforts since the 1980s. Efforts have centred around removal of an invasive European beach grass species, Ammophila arenaria, introduced in the early 1900s to stabilize the dunes to protect landward communities from coastal flooding and storm surges. Despite effectively stabilizing the foredune, A. arenaria forms monotypic vegetation stands, with highly dense roots, rhizomes, and above-ground biomass that can lead to pronounced scarping of the seaward slope, alongshore steering of wind and sediment, a lack of landward transfer of sand, and a steeper, more peaked profile. Effective foredune restoration must consider the coupled interactions between dominant plant type and the geomorphic processes that influence dune form. A 5 ha reach of recently restored foredune was monitored biannually with terrestrial laser scanner and uncrewed aerial systems platforms between 2015 and 2021 to characterize the impacts of dynamic restoration on foredune form and resiliency. This reach included two control plots: (1) native, non-restored and (2) invasive, and three restored plots revegetated with native species: (3) a native grass (Elymus mollis), (4) a low-lying herb and subshrub assemblage, and (5) a mixture of the native grass, herbs, and subshrubs. After five growing seasons, restored plots exhibited distinct geomorphic and sediment budget differences. Natively vegetated plots recovered from extensive scarping 2 years faster than the invasive plot. Restored plots saw foredune height (0.5–0.7 m) and width increase, landward extension (1 m) while maintaining a similar seaward position, and positive lee-slope sediment budgets that exceeded both control plots. These results suggest that the native vegetation plots allowed increased landward sand transport across the foredune, and increased the capacity of the foredune to recover more quickly following dune scarping

Suicidal Presentations to Emergency Departments in a Large Australian Public Health Service over 10 Years

This paper presents trends and characteristics for 32,094 suicidal presentations to two Emergency Departments (EDs) in a large health service in Australia across a 10-year period (2009&ndash;2018). Prevalence of annual suicidal presentations and for selected groups of consumers (by sex, age groups, and ethnicity) was determined from a machine learning diagnostic algorithm developed for this purpose and a Bayesian estimation approach. A linear increase in the number of suicidal presentations over 10 years was observed, which was 2.8-times higher than the increase noted in all ED presentations and 6.1-times higher than the increase in the population size. Females had higher presentation rates than males, particularly among younger age groups. The highest rates of presentations were by persons aged 15&ndash;24. Overseas-born persons had around half the rates of suicidal presentations than Australian-born persons, and Indigenous persons had 2.9-times higher rates than non-Indigenous persons. Of all presenters, 70.6% presented once, but 5.7% had five or more presentations. Seasonal distribution of presentations showed a peak at the end of spring and a decline in winter months. These findings can inform the allocation of health resources and guide the development of suicide prevention strategies for people presenting to hospitals in suicidal crisis