A mathematica‐based CAL matrix‐theory tutor for scientists and engineers

Under the TLTP initiative, the Mathematics Departments at Imperial College and Leeds University are jointly developing a CAL method directed at supplementing the level of mathematics of students entering science and engineering courses from diverse A‐level (or equivalent) backgrounds. The aim of the joint project is to maintain — even increase ‐ the number of students enrolling on such first‐year courses without lowering the courses’ existing mathematical standards

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

Adaptive finite element simulation of three-dimensional surface tension dominated free-surface flow problems

An arbitrary Lagrangian--Eulerian finite element method is described for the solution of time-dependent, three-dimensional, free-surface flow problems. Many flows of practical significance involve contact lines, where the free surface meets a solid boundary. This contact line may be pinned to a particular part of the solid but is more typically free to slide in a manner that is characterised by the dynamic contact angle formed by the fluid. We focus on the latter case and use a model that admits spatial variation of the contact angle: thus permitting variable wetting properties to be simulated. The problems are driven by the motion of the fluid free surface (under the action of surface tension and external forces such as gravity) hence the geometry evolves as part of the solution, and mesh adaptivity is required to maintain the quality of the computational mesh for the physical domain. Continuous mesh adaptivity, in the form of a pseudo-elastic mesh movement scheme, is used to move the interior mesh nodes in response to the motion of the fluid's free surface. Periodic, discrete remeshing stages are also used for cases in which the fluid volume has grown, or is sufficiently distorted, by the free-surface motion. Examples are given of a droplet sliding on an inclined uniform plane and of a droplet spreading on a surface with variable wetting properties

Integral equation analysis of the driven-cavity boundary singularity

AbstractA specially-modified boundary integral equation (BIE) method is used to investigate the viability of the singular boundary conditions of the well known driven-cavity Stokes flow problem, a bench-mark problem of computational fluid dynamics. We introduce small ‘leaks’ to replace the singularities, thus creating a perturbed, physically realizable problem. We make two discoveries, namely: (i) unexpectedly, the introduction of the leaks affects the flow field at considerably greater distances from the leaks than one might perhaps intuitively predict; and (ii) the full, numerical BIE solution reveals that the far field, asymptotic, closed-form solution for the flowfield of the perturbed problem is a surprisingly accurate representation of the flow even in the near field

Motion of nanodroplets near edges and wedges

Nanodroplets residing near wedges or edges of solid substrates exhibit a disjoining pressure induced dynamics. Our nanoscale hydrodynamic calculations reveal that non-volatile droplets are attracted or repelled from edges or wedges depending on details of the corresponding laterally varying disjoining pressure generated, e.g., by a possible surface coating.Comment: 12 pages, 7 figure

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

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