An exponentially convergent Volterra-Fredholm method for integro-differential equations

Extending the authors’ recent work [15] on the explicit computation of error bounds for Nystrom solvers applied to one-dimensional Fredholm integro-differential equations (FIDEs), presented herein is a study of the errors incurred by first transforming (as in, e.g., [21]) the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE). The VFIE is solved via a novel approach that utilises N-node Gauss-Legendre interpolation and quadrature for its Volterra and Fredholm components respectively: this results in numerical solutions whose error converges to zero exponentially with N, the rate of convergence being confirmed via large- N asymptotics. Not only is the exponential rate inherently far superior to the algebraic rate achieved in [21], but also it is demonstrated, via diverse test problems, to improve dramatically on even the exponential rate achieved in [15] via direct Nystrom discretisation of the original FIDE; this improvement is confirmed theoretically

A priori Nyström-method error bounds in approximate solutions of 1-D Fredholm integro-differential equations

A novel procedure is proposed for the a priori computation of error bounds for the ubiquitous Nyström solver applied to one-dimensional Fredholm integro-differential equations. The distinctive feature of the new approach is that the bounds are computed not only to spectral accuracy, but also explicitly, and in terms of only the numerical solution itself. Details are given of both the error analysis and its numerical implementation, and a corroborative asymptotic theory is developed in order to yield independent predictions of the convergence rates expected from Nyström discretisations of increasing order. All theory is first convincingly validated on a proof-of-concept continuous-kernel test problem whose solution is a priori known. The method is then applied to a novel integro-differential-equation formulation of a static, fourth-order, Euler-Bernoulli beam-deflection boundary-value problem in which the flexural rigidity varies along the beam, and for which no exact solution is attainable; in this case, validation of the resulting discontinuous-kernel approach is achieved using an asymptotic solution derived on the (realistic) assumption that variations in the cross-section of the beam occur on spatial scales an order of magnitude less than the beam’s length and width. Potential limitations of the new approach are discussed

Error analysis of a spectrally accurate Volterra-transformation method for solving 1-D Fredholm integro-differential equations

Spectrally accurate a priori error estimates for Nyström-method approximate solutions of one-dimensional Fredholm integro-differential equations (FIDEs) are obtained indirectly by transforming the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE), which is solved via a novel approach that utilises N-node Gauss-Legendre interpolation and quadrature for its Volterra and Fredholm components respectively. Errors in the numerical solutions of the VFIE converge to zero exponentially with N, the rate of convergence being confirmed via large-N asymptotics. Not only is the exponential rate far superior to the algebraic rate achieved in previous literature [29] but also it is demonstrated, via diverse test problems, to improve dramatically on even the exponential rate achieved in the approach [21] of direct Nyström discretisation of the original FIDE; this improvement is confirmed theoretically

Variational Coupling of Wave Slamming against Elastic Masts

We present a novel approach to fluid-structure interactions (FSI) that preserves energy and phase-space structure owing to the variational and Hamiltonian techniques used. We posit a variational principle (VP), for nonlinear potential-flow wave dynamics coupled to a nonlinear hyperelastic mast, and derive its linearization. Both linear and nonlinear formulations can then be discretized in a classical-mechanical VP, using nite element expansions

A Cost-Effectiveness Protocol for Flood-Mitigation Plans Based on Leeds’ Boxing Day 2015 Floods

Inspired by the Boxing Day 2015 flood of the River Aire in Leeds, UK, and subsequent attempts to mitigate adverse consequences of flooding, the goals considered are: (i) to revisit the concept of flood-excess volume (FEV) as a complementary diagnostic for classifying flood events; (ii) to establish a new roadmap/protocol for assessing flood-mitigation schemes using FEV; and, (iii) to provide a clear, graphical cost-effectiveness analysis of flood mitigation, exemplified for a hypothetical scheme partially based on actual plans. We revisit the FEV concept and present it as a three-panel graph using thresholds and errors. By re-expressing FEV as a 2m -deep square lake of equivalent capacity, one can visualise its dimensions in comparison with the river valley considered. Cost-effectiveness of flood-mitigation measures is expressed within the FEV square-lake; different scenarios of our hypothetical flood-mitigation scheme are then presented and assessed graphically, with each scenario involving a combination, near and further upstream of Leeds, of higher (than existing) flood-defence walls, enhanced flood-plain storage sites, giving-room-to-the-river bed-widening and natural flood management. Our cost-effectiveness analysis is intended as a protocol to compare and choose between flood-mitigation scenarios in a quantifiable and visual manner, thereby offering better prospects of being understood by a wide audience, including citizens and city-council planners. Using techniques of data analysis combined with general river hydraulics, common-sense and upper-bound estimation, we offer an accessible check of flood-mitigation plans

A perspective on SIDS pathogenesis. The hypotheses: plausibility and evidence

Several theories of the underlying mechanisms of Sudden Infant Death Syndrome (SIDS) have been proposed. These theories have born relatively narrow beach-head research programs attracting generous research funding sustained for many years at expense to the public purse. This perspective endeavors to critically examine the evidence and bases of these theories and determine their plausibility; and questions whether or not a safe and reasoned hypothesis lies at their foundation. The Opinion sets specific criteria by asking the following questions: 1. Does the hypothesis take into account the key pathological findings in SIDS? 2. Is the hypothesis congruent with the key epidemiological risk factors? 3. Does it link 1 and 2? Falling short of any one of these answers, by inference, would imply insufficient grounds for a sustainable hypothesis. Some of the hypotheses overlap, for instance, notional respiratory failure may encompass apnea, prone sleep position, and asphyxia which may be seen to be linked to co-sleeping. For the purposes of this paper, each element will be assessed on the above criteria

Spectrally accurate Nyström-solver error bounds for 1-D Fredholm integral equations of the second kind

We present the theory underlying and computational implementation of analytical predictions of error bounds for the approximate solution of one-dimensional Fredholm integral equations of the second kind. Through asymptotic estimates of near-supremal operator norms, readily implementable formulae for the error bounds are computed explicitly using only the numerical solution of Nyström-based methods on distributions of nodes at the roots or extrema of diverse orthogonal polynomials. Despite the predicted bounds demanding no a priori information about the exact solution, they are validated to be spectrally accurate upon comparison with the explicit computational error accruing from the numerical solution of a variety of test problems, some chosen to be challenging to approximation methods, with known solutions. Potential limitations of the theory are discussed, but these are shown not to arise in the numerical computations

Computable theoretical error bounds for Nyström methods for 1-D Fredholm integral equations of the second kind

New expressions for computable error bounds are derived for Nyström method approximate solutions of one-dimensional second-kind Fredholm integral equations. The bounds are computed using only the numerical solution, and so require no a priori knowledge of the exact solution. The analysis is implemented on test problems with both well-behaved and “Runge-phenomenon” solutions, and the computed predictions are shown to be in impressive quantitative agreement with the true errors obtained from known exact solutions of the test problems. For independent computational validation, both Lagrange and barycentric interpolation are employed on grids with both regularly spaced nodes and those located at the roots or extrema of orthogonal polynomials. For independent theoretical validation, asymptotic estimates are derived for the convergence rates of the observed computational errors

Cost-effective simulation and prediction of explosions for military and public safety, and for improved oil extraction

An MoD-funded research programme based in Applied Mathematics at Leeds University has resulted in demonstrable long-term and ongoing benefits on diverse fronts for beneficiaries in a range of public and private sectors. First, by guaranteeing robustness and reliability of bespoke numerical methods for the MoD, the joint research led to substantial financial savings in ballistic-development programmes, thereby enabling the delivery of advanced research output cost-effectively under severe budgetary pressures. As a result, QinetiQ was placed as a world leader in the simulation of explosions, which supported the MoD to rapidly assess and develop countermeasures to the ever-changing threats faced by British Forces in Afghanistan and Iraq, and to reduce casualties. It also enabled government agencies to assess threats to transport and public-building infrastructure. Second, the joint research underpinned substantial recurrent income for QinetiQ, who has additionally developed the codes with the oil industry to develop a new explosive perforator for oil extraction that has not only led to demonstrable improvements in both extraction efficiency and research-and-development costs, but has also yielded recurrent licensing royalties

Inertially Induced Cyclic Solutions in Thin-Film Free-Surface Flows

New mechanisms are discovered regarding the effects of inertia in the transient Moffatt–Pukhnachov problem (J. Méc., vol. 187, 1977, pp. 651–673) on the evolution of the free surface of a viscous film coating the exterior of a rotating horizontal cylinder. Assuming two-dimensional evolution of the film thickness (i.e. neglecting variation in the axial direction), a multiple-timescale procedure is used to obtain explicitly parameterized high-order asymptotic approximations of solutions of the spatio-temporal evolution equation. Novel, hitherto-unexplained transitions from stability to instability are observed as inertia is increased. In particular, a critical Reynolds number Rec is predicted at which occurs a supercritical pitchfork bifurcation in wave amplitude that is fully explained by the new asymptotic theory. For Re Rec, stable temporally periodic solutions with leading-order amplitudes proportional to .Re Rec/1=2 are found, i.e. in the régime in which previous related literature predicts exponentially divergent instability. For ReDRec, stable solutions are found that decay algebraically to a steady state. A model solution is proposed that not only captures qualitatively the interaction between fundamental and higher-order wave modes but also offers an explanation for the formation of the lobes observed in Moffatt’s original experiments. All asymptotic theory is convincingly corroborated by numerical integrations that are spectrally accurate in space and eighth/ninth-order accurate in time