A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domainswith piecewise smooth boundaries is presented. It is intendedfor problems describing the Stokes flow in the situations whereone has corners or cusps formed by parts of the domain boundaryand, due to the nature of the boundary conditions on these partsof the boundary, these regions have a global effect on the shapeof the whole domain and hence have to be resolved with sufficientaccuracy. The algorithm combines the boundary integral equationmethod for the main part of the flow domain and the finite-elementmethod which is used to resolve the corner/cusp regions. Twoparts of the solution are matched along a numerical ‘internalinterface’ or, as a variant, two interfaces, and theyare determined simultaneously by inverting a combined matrixin the course of iterations. The algorithm is illustrated byconsidering the flow configuration of ‘curtain coating’,a flow where a sheet of liquid impinges onto a moving solidsubstrate, which is particularly sensitive to what happens inthe corner region formed, physically, by the free surface andthe solid boundary. The ‘moving contact line problem’is addressed in the framework of an earlier developed interfaceformation model which treats the dynamic contact angle as partof the solution, as opposed to it being a prescribed functionof the contact line speed, as in the so-called ‘slip models’.     

Some exact results for nonlinear diffusion with absorption

Simple exact solutions and first integrals are obtained fornonlinear diffusion incorporating absorption. These are obtainedby the standard techniques of separation of variables and theuse of invariant one-parameter group transforma-tions to reducethe governing partial differential equation to various ordinarydifferential equations. For two of the equations so obtained,first integrals are deduced which subsequently give rise toa number of explicit simple solutions. As with all special solutionsof nonlinear partial differential equations, the associatedinitial and boundary conditions are imposed by the particularfunctional form of the solution and irrespective of their physicalapplicability, simple exact solutions are always important,because one of the key features of nonlinearity is the rangeand variety of response which is often bizarre and unexpected,but which is frequently embodied in the simplest of exact solutions.Many of the solutions obtained here are illustrated graphicallywith particular reference to the phenomena of ‘extinction’and ‘blow-up’ and in general demonstrate a widevariety of differing physical response embodied in the disposableconstants, which is characteristic of nonlinear theories.     

Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries

In the paper we carry out a complete analysis of several efficientnumerical methods for the solution of boundary integral equationsdefined on a non-smooth boundary. In particular the solutionof the Helmholtz equation in the exterior of a closed wedgeis studied. The analytical behaviour of the solution of theresulting boundary integral equation (with a non-compact operator)near the wedge is investigated. Numerical analysis of the collocationand iterated collocation method for the problem is presented.Graded meshes are used to reflect the ‘singular’behaviour of the analytical solution, as well as the degreeof the polynomial approximant, in order to yield results with‘optimal convergence rates’. Finally the convergenceanalysis of some modified two-grid iterative methods for thefast solution of the resulting linear systems is given and numericalresults are presented which agree with the theoretical predictions.     

Modelling the growth of soil-borne fungi in response to carbon and nitrogen

Email: Angelique.Lamour{at}medew.fyto.wau.nl Growth of soil-borne fungi is poorly described and understood,largely because non-destructive observations on hyphae in soilare difficult to make. Mathematical modelling can help in theunderstanding of fungal growth. Except for a model by Paustian& Schnürer (1987a), fungal growth models do not considercarbon and nitrogen contents of the supplied substrate, althoughthese nutrients have considerable effects on hyphal extensionin soil. We introduce a fungal growth model in relation to soilorganic matter decomposition dealing with the detailed dynamicsof carbon and nitrogen. Substrate with a certain carbon: nitrogenratio is supplied at a constant rate, broken down and then takenup by fungal mycelium. The nutrients are first stored internallyin metabolic pools and then incorporated into structural fungalbiomass. Standard mathematical procedures were used to obtainoverall-steady states of the variables (implicitly from a cubicequation) and the conditions for existence. Numerical computationsfor a wide range of parameter combinations show that at mostone solution for the steady state is biologically meaningful,specified by the conditions for existence. These conditionsspecify a constraint, namely that the ‘energy’ (interms of carbon) invested in breakdown of substrate should beless than the ‘energy’ resulting from breakdownof substrate, leading to a positive carbon balance. The biologicalinterpretation of the conditions for existence is that for growththe ‘energy’ necessary for production of structuralfungal biomass and for maintenance should be less than the mentionedpositive carbon balance in the situation where all substrateis colonized. In summary, the analysis of this complicated fungalgrowth model gave results with a clear biological interpretation.     

Radial Limits of Bloch Functions in the unit Disc

Although a function in the Bloch space may have no radial limits,it is shown that there exist bounded linear functionals whichgive ‘average radial limits’ over an interval onthe boundary. An ‘abelian–tauberian’ theoremis proved, characterizing the existence of a radial limit ata given boundary point in terms of these functionals.     

Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

Discretization of autonomous ordinary differential equationsby numerical methods might, for certain step sizes, generatesolution sequences not corresponding to the underlying flow—so-called‘spurious solutions’ or ‘ghost solutions’.In this paper we explain this phenomenon for the case of explicitRunge-Kutta methods by application of bifurcation theory fordiscrete dynamical systems. An important tool in our analysisis the domain of absolute stability, resulting from the applicationof the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inheritedby the difference scheme induced by the numerical method whilethe stability type of these inherited genuine fixed points iscompletely determined by the method's domain of absolute stability.We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the correspondinglimit step size. Moreover, we show in which way the bifurcationsoccurring at the limit step size are connected to the valuesof the stability function on the boundary of the domain of absolutestability, where we pay special attention to bifurcations leadingto spurious solutions. In order to explain a certain kind ofspurious fixed points which are not connected to the set ofgenuine fixed points, we interprete the domain of absolute stabilityas a Mandeibrot set and generalize this approach to nonlinearproblems.     

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

Planar elasticity problems are considered for thin domains fixedalong a small part of the end region boundary. The analysisinvolves two small parameters: the normalized thickness of thebody and the normalized length of the fixed part of the boundary.The aim of the paper is to derive an asymptotic approximationof the solution to a boundary-value problem in such a domainand, in particular, analyze the ‘effective boundary conditions’,which occur for the leading-order terms of the asymptotics.We include applications for problems of both anti-plane shearand plane strain elasticity.     

Convergence estimates for the numerical approximation of homoclinic solutions

Received on 21 November 1995. Revised on 12 July 1996. This article is concerned with the numerical computation ofhomoclinic solutions converging to a hyperbolic or semi-hyperbolicequilibrium of a system u = f(u, µ). The approximationis done by replacing the original problem with a boundary valueproblem on a finite interval and introducing an additional phasecondition to make the solution unique. Numerical experimentshave indicated that the parameter µ is much better approximatedthan the homoclinic solution. This was proved in Schecter (1995IMA J. Numer Anal. 15, 23–60) for phase conditions satisfyingan additional ‘niceness’ assumption, which is unfortunatelynot satisfied for the phase condition most commonly used innumerical experiments and which actually suggested the super-convergenceresult. Here, this result is proved for arbitrary phase conditions.Moreover, it is shown that it suffices to approximate the originalboundary value problem to first order when considering semi-hyperbolicequilibria, extending a result of Schecter (1993 SIAM J. NumerAnal. 30, 1155–78). Permanent address: WIAS, MohrenstraBe 39, 10117 Berlin, Germany     

On the Robin problem in fractal domains

We study the solution to the Robin boundary problem for theLaplacian in a Euclidean domain. We present some families offractal domains where the infimum of the solution to the mixedDirichlet–Robin boundary problem is greater than 0, andsome other families of domains where it is equal to 0. We alsogive a new result on ‘trap domains’ defined in Burdzy,Chen and Marshall (Math. Z.), that is, domains where reflectingBrownian motion takes a long time to reach the center of thedomain.     

Systems with Time Delay in the Calculus of Variations: The Method of Steps

In a recent article (see [8]), we derived necessary and sufficientconditions for minima for the fixed-endpoint problem in thecalculus of variations involving a constant delay in the phasecoordinates. These conditions are expressed, explicitly, interms of the first and second variations. The vanishing of thefirst variation is characterized in terms of an extended Euler'sequation, just as for delay-free problems, but the characterizationof the conditions on the second variation remained unsolved.In this paper we convert, through the ‘method of steps’,the delay problem into one without delay. Although this problemwill not have fixed-endpoint constraints, it allows us to introduce,in a natural way, the concept of ‘conjugate sequence’;this solves the main difficulty for delay problems, namely,to give initial conditions for existence and uniqueness of solutionsof the Hamiltonian system (which is a difference–differentialsystem with both advanced and retarded arguments). The conditionson the second variation are then characterized by an extra conditionwhich is based exclusively on a solution of a given matrix Riccatiequation.     

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Variational ‘self-consistent’ estimates for nonlinearproblems are formulated, building on a variational formulationpreviously developed by the authors. The formulation employsa linear ‘comparison medium’ for whose propertiessome ‘self-consistent’ choice is made. In contrastto linear problems, three possible self-consistent choices presentthemselves. The results that they give are analysed for twoparticular systems (a nonlinear dielectric and a nonlinear lossycomposite) for which bounds are already available. Estimatesbased on self-consistent embedding of a single inclusion ina homogeneous matrix composed of ‘comparison material’are also developed.     

Steady-state motion of a mode-III crack on imperfect interfaces

The aim of the present paper is to analyse the behaviour ofthe stress and displacement fields in the vicinity of the tipof a crack moving along a bi-material interface. For simplicity,we consider a straight interface of infinite extent. We assumethat the two phases are separated by a thin layer which is either‘soft’ or ‘stiff’ compared to the othertwo phases. We derive the transmission conditions which takeinto account the material properties of the layer and modelthe way the load is transferred across the layer from one phaseto the other. We assume that the point of interchange in theboundary/transmission conditions coincides with the crack tipthat moves along the interface boundary with a constant speed.We develop an integral equation formulation and derive asymptoticformulae for the out-of-plane displacement and the Mode-IIIstress intensity factor associated with such a motion of thecrack inside the interphase layer. The theoretical results areillustrated by numerical examples.     

The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory

The aim of this paper is to develop a straightforward analysisof the Galerkin method for two-dimensional boundary integralequations of the first kind with logarithmic kernels. A distinctivefeature of the analysis is that no appeal is made to ‘coercivity’,as a result of which some existence questions cannot be answereddirectly. In return, however, the analysis has no special difficultyin handling corners, cusps, or open arcs. Instead of coercivity,the central feature of the analysis is the positive-definiteproperty of the integral operator for small enough contours.Rates of convergence are predicted theoretically and, in particular,certain linear functionals are shown to exhibit ‘superconvergence’.Numerical results supporting the theory are given in the companionpaper Sloan & Spence (1987) for problems on both open andclosed polygonal arcs.     

Tollmien-Schlichting/vortex interactions in compressible boundary-layer flows

A weakly nonlinear interaction of oblique Tollmien-Schlichtingwaves and longitudinal vortices in compressible, high Reynoldsnumber, boundary-layer flow over a flat plate is consideredfor all ranges of the Mach number. The interaction equationsconsist of equations for the vortex which is indirectly forcedby the waves via a boundary condition, whereas a vortex termappears in the amplitude equation for the wave pressure. Thedownstream solution properties of interaction equations arefound to depend on the sign of an interaction coefficient. Thisparticular type of weakly nonlinear interaction was first proposedby Hall & Smith (1989), who considered incompressible flows;however, there are some errors in their formulation. Correctedresults for the incompressible regime are presented for comparisonwith those calculated for compressible flows. Compressibilityis found to have a significant effect on the interaction properties,principally through its impact on the waves and their governingmechanism, the ‘triple-deck’ structure. It is foundthat, in general, the flow quantities will grow slowly withincreasing downstream coordinate. However, for flows with Machnumber values below 2, there exists a small band of wave obliquenessangles for which the solutions terminate in abrupt, finite-distance‘break-ups’.     

A Globally Convergent Version of REQP for Constrained Minimization

This paper seeks to improve the theoretical basis for the nonlinearprogramming algorithm REQP. A modification to the standard formof the method is presented which allows global convergence tobe proved under assumptions that are more realistic than thosepreviously required. Essentially, the modification involvesan additional test after each step to determine whether satisfactoryprogress is being made. Reduction in the penalty parameter canonly take place after a step that is judged to be ‘successful’.Numerical examples are included which show how this modificationcan help to protect the method from being ‘trapped’close to a constraint at a nonoptimal point.     

Modeling weakly nonlinear acoustic wave propagation

Three weakly nonlinear models of lossless, compressible fluidflow—a straightforward weakly nonlinear equation (WNE),the inviscid Kuznetsov equation (IKE) and the Lighthill–Westerveltequation (LWE)—are derived from first principles and theirrelationship to each other is established. Through a numericalstudy of the blow-up of acceleration waves, the weakly nonlinearequations are compared to the ‘exact’ Euler equations,and the ranges of applicability of the approximate models areassessed. By reformulating these equations as hyperbolic systemsof conservation laws, we are able to employ a Godunov-type finite-differencescheme to obtain numerical solutions of the approximate modelsfor times beyond the instant of blow-up (that is, shock formation),for both density and velocity boundary conditions. Our studyreveals that the straightforward WNE gives the best results,followed by the IKE, with the LWE's performance being the poorestoverall.     

A queueing model of activities in an intensive care unit

^** Email: griffiths{at}cardiff.ac.uk Activities in an intensive care unit (ICU) at a major teachinghospital are modelled by means of a queue-theoretic approach.Using data supplied by the ICU relating to the admissions process,the bed availability and the length of stay of patients, itwas possible to fit theoretical distributions to the observed‘arrival’ and ‘service’ distributions.Queueing equations relevant to a multi-channel system havingrandom arrivals and hyper-exponential service times for eachchannel are set up, and solved iteratively. Results obtainedmatch well with observations, and the model is then utilisedto investigate several ‘what if? ’ scenarios. Referenceis made to a simulation model developed in conjunction withthe queueing model.     

Decoupling of bidiagonal systems involving singular blocks

For one-step difference equations, where the matrix coefficientsmay be singular, a stability analysis based on using fundamentalsolutions and their inverses does not apply. This paper showshow well-boundedness of the Green's function leads to a kindof dichotomy of the fundamental solution, including certain‘parasitic solutions’ (which arise because of thesingularity of the fundamental solutions). This then is usedto show how one can find a stable decoupling and thus a numericalalgorithm for solving a discrete boundary-value problem. Severalexamples sustain the analysis.