On the convergence of spectral deferred correction methods

In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this high-order ordinary differential equation (ODE) solver is typically described wherein a low-order approximation (such as forward or backward Euler) is lifted to higher order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard integral equation solver. In doing so, our chief finding is that it is not the low-order solver that picks up the order of accuracy with each correction, but it is the underlying quadrature rule of the right hand side function that is solely responsible for picking up additional orders of accuracy. Our proofs point to a total of three sources of errors that SDC methods carry: the error at the current time point, the error from the previous iterate, and the numerical integration error that comes from the total number of quadrature nodes used for integration. The second of these two sources of errors is what separates SDC methods from Picard integral equation methods; our findings indicate that as long as difference between the current and previous iterate always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying "solver" is inconsistent the underlying ODE. From this vantage, we solidify the prospects of extending spectral deferred correction methods to a larger class of solvers to which we present some examples.Comment: 29 page

General practice registrars’ intentions for future practice: Implications for rural medical workforce planning

The models of practice that general practice registrars (GPRs) envisage undertaking will affect workforce supply. The aim of this research was to determine practice intentions of current GPRs in a regional general practice training program (Coast City Country General Practice Training). Questionnaires were circulated to 220 GPRs undertaking general practice placements to determine characteristics of ideal practice models and intentions for future practice. Responses were received for 99 participants (45%). Current GPRs intend to work an average of less than eight half-day sessions/week, with male participants intending to work more hours (t(91) = 3.528, P = 0.001). More than one-third of this regional cohort intends to practice in metropolitan centres. Proximity to family and friends was the most important factor influencing the choice of practice location. Men ranked remuneration for work as more important (t (88) = –4.280, P \u3c 0.001) and women ranked the ability to work part-time higher (t(94) = 3.697, P \u3c 0.001). Fee-for-service payment alone, or in combination with capitation, was the preferred payment system. Only 22% of Australian medical graduates intend to own their own practice compared with 52% of international medical graduates (χ2(1) = 8.498, P = 0.004). Future general practitioners (GPs) intend to work fewer hours than current GPs. Assumptions about lifestyle factors, practice models and possible professional roles should be carefully evaluated when developing strategies to recruit GPs and GPRs into rural practice

Constructive Upheaval: Railway Labor Executives\u27Ass\u27n v. Gibbons and the Choice of Clause Challenge to Traditions of Statutory Construction

When confronted with constitutional challenges toCongress\u27s legislative authority, courts must build theiranalyses on an interpretation of the statute\u27s language.Such cases implicate principles of statutory constructionthat lay the groundwork for the rulings that follow.Throughout American judicial history, courts havefavored flexible interpretation to protect Congress\u27senactments from constitutional attack. The SupremeCourt\u27s decision in Railway Labor Executives\u27 Associationv. Gibbons dramatically departed from that tradition,suggesting instead that legislation should be categorizedas a particular type of law to ensure that Congress doesnot overstep the boundaries of its enumerated powers.Although its shift in perspective took time to register,Gibbons\u27s novel approach, here referred to as a choice ofclause analysis, introduced an alternative to flexibleconstruction with the potential to shake the foundations offuture constitutional cases. Inspired by a recent cycle ofcases that confronted Gibbons\u27s reasoning, this Noteexamines the choice of clause analysis and attempts tounderstand its place among the canons of statutoryconstruction. Building on the fundamental differencesbetween the choice of clause analysis and a long-followedprinciple of construction, this Note argues that the choiceof clause analysis should not be given a role in the processof statutory construction because of the potential forGibbons\u27s approach to dramatically reshape the outcomesof constitutional cases