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Various numerical methods have been proposed for the solution of weakly singular Volterra integral equations but, for the most part, authors have dealt with linear or one-dimensional weakly singular Volterra integral equations, or have assumed that these equations have smooth solutions. The main purpose of this paper is to propose and analyse a numerical method for the solution of two-dimensional nonlinear weakly singular Volterra integral equations of the second kind. In general the solutions of these equations exhibit singularities in their derivatives at t = 0 even if the forcing functions are smooth. To overcome these difficulties a simple smoothing change of variables is proposed. By applying this transformation an equation is obtained which, while still being weakly singular, can have a solution as smooth as is required. We then solve this transformed integral equation using Navot’s quadrature rule for computing integrals with an end point singularity. A new extension of a discrete Gronwall inequality allows us to prove convergence and obtain an error estimate. The theoretical results are then verified by numerical examples.

Our aim is to simulate a nonlinear system of ODEs describing the flow in smectic liquid crystals. The nonlinear system is first linearized. We present a direct approach to compute the exact analytic solution of this linear system and use this solution as a starting profile in the Matlab package bvpsuite2.0 to obtain the approximate solution to the nonlinear system. Although, the solution of the nonlinear system has steep boundary layers and therefore is difficult to resolve, we demonstrate that bvpsuite2.0 can cope with the problem and provide an approximation with reasonable accuracy.

In this paper, we discuss two-level implicit methods for the solution of a special type of fourth order parabolic partial differential equation of the form uxxxx-2uxxt+utt=f(x,t,u), 00 subject to appropriate initial and Dirichlet boundary conditions by converting the original problem to a coupled system of two second order parabolic equations. We use only three spatial grid points and it is not required to discretize the boundary conditions. The proposed Crank-Nicolson type scheme is second order accurate in both the temporal and spatial dimensions while the compact Crandall's type scheme is second order accurate in temporal and fourth order accurate in spatial dimension. The methods do not require any fictitious nodes outside the solution domain for handling the boundary conditions. For a fixed mesh ratio parameter (t/x2), the proposed Crandall's type method behaves like a fourth order method in space. Using matrix stability analysis, the proposed methods are shown to be unconditionally stable. The resulting implicit difference formulas gives block tri-diaginal matrix structure which is solved efficiently using block Gauss-Seidel method or block Newton method depending on linear or nonlinear behaviour of the equations. The methods compute the numerical value of u and time-dependent Laplacian uxxut, simultaneously. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed methods.

A polynomial collocation method is proposed for the numerical solution of a class of singular integro-differential equations of Cauchy type; the collocation points are chosen to be the Chebyshev nodes. Function spaces are defined and theorems concerning the boundedness of certain operators are developed. Convergence of the numerical method is demonstrated in weighted uniform normed spaces of continuous functions; convergence rates are then determined in accordance with the smoothness of the functions characterizing the problem. Numerical examples are provided which go some way to confirming these estimates.

Mixed Volterra–Fredholm integral equations can arise from mathematical modelling of the spatio-temporal development of an epidemic. In this paper a hybrid Legendre block-pulse method is used to provide solutions to a general integral equation of this type. The main idea of this approach is to reduce a Volterra–Fredholm integral equation to a system of algebraic equations. An order convergence analysis is provided. The accuracy of the proposed method is then confirmed by solving some problems, the first being a nonlinear epidemic problem.

While studying a problem in biomedical research a simple diffusion problem arose which admitted a solution by Fourier transforms. It was natural to ask if the same problem could be solved by Laplace transforms. In this note, we provide three solution techniques using Laplace transforms, with the last leading to a number of novel mathematical identities.

We present two novel two-step explicit methods for the numerical solution of the second order initial value problem on a variable mesh. In the case of a constant mesh the method is superstable in the sense of Chawla (1985). Numerical experimentation is provided to verify the stability analysis.

In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved.

This article examines the steady flow of a smectic A liquid crystal sample that is initially aligned in a classical "bookshelf" geometry confined between parallel plates and is then subjected to a lateral pressure gradient which is perpendicular to the initial local smectic layer arrangement. The nonlinear dynamic equations are derived. These equations can be linearized and solved exactly to reveal two characteristic length scales that can be identified in terms of the material parameters and reflect the boundary layer behavior of the velocity and the director and smectic layer normal orientations. The asymptotic properties of the nonlinear equations are then investigated to find that these length scales apparently manifest themselves in various aspects of the solutions to the nonlinear steady state equations, especially in the separation between the orientations of the director and smectic layer normal. Non-Newtonian plug-like flow occurs and the solutions for the director profile and smectic layer normal share features identified elsewhere in static liquid crystal configurations. Comparisons with numerical solutions of the nonlinear equations are also made.

A new integration technique, which is suitable for integrands with multiple weak singularities, is introduced. Local truncation errors are given. This scheme, when applied to the Beta function, is shown to emerge naturally from discrete fractional integration. To illustrate the effectiveness of the integration method a numerical example is provided, with somewhat unexpected convergence results.

This paper is concerned with an overview of upwinding schemes, and further nonlinear applications of a recently introduced high resolution upwind differencing scheme, namely the ADBQUICKEST [V.G. Ferreira, F.A. Kurokawa, R.A.B. Queiroz, M.K. Kaibara, C.M. Oishi, J.A. Cuminato, A.F. Castelo, M.F. Tome, S. McKee, assessment of a high-order finite difference upwind scheme for the simulation of convection–diffusion problems, International Journal for Numerical Methods in Fluids 60 (2009) 1–26]. The ADBQUICKEST scheme is a new TVD version of the QUICKEST [B.P. Leonard, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Computer Methods in Applied Mechanics and Engineering 19 (1979) 59–98] for solving nonlinear balance laws. The scheme is based on the concept of NV and TVD formalisms and satisfies a convective boundedness criterion. The accuracy of the scheme is compared with other popularly used convective upwinding schemes (see, for example, Roe (1985) [19] , Van Leer (1974) [18] and Arora & Roe (1997) [17] ) for solving nonlinear conservation laws (for example, Buckley–Leverett, shallow water and Euler equations). The ADBQUICKEST scheme is then used to solve six types of fluid flow problems of increasing complexity: namely, 2D aerosol filtration by fibrous filters; axisymmetric flow in a tubular membrane; 2D two-phase flow in a fluidized bed; 2D compressible Orszag–Tang MHD vortex; axisymmetric jet onto a flat surface at low Reynolds number and full 3D incompressible flows involving moving free surfaces. The numerical simulations indicate that this convective upwinding scheme is a good generic alternative for solving complex fluid dynamics problems.

SUMMARY This work describes a methodology to simulate free surface incompressible multiphase flows. This novel methodology allows the simulation of multiphase flows with an arbitrary number of phases, each of them having different densities and viscosities. Surface and interfacial tension effects are also included. The numerical technique is based on the GENSMAC front-tracking method. The velocity field is computed using a finite-difference discretization of a modification of the Navier–Stokes equations. These equations together with the continuity equation are solved for the two-dimensional multiphase flows, with different densities and viscosities in the different phases. The governing equations are solved on a regular Eulerian grid, and a Lagrangian mesh is employed to track free surfaces and interfaces. The method is validated by comparing numerical with analytic results for a number of simple problems; it was also employed to simulate complex problems for which no analytic solutions are available. The method presented in this paper has been shown to be robust and computationally efficient. Copyright © 2012 John Wiley & Sons, Ltd.

A mathematical model in the form of two coupled diffusion equations is provided for a competitive chemical reaction between an antigen and a labeled antigen for antibody sites on a cell wall; boundary conditions are such that the problem is both nonlinear and nonlocal. This is then recharacterized first as a pair of coupled singular integro-differential equations and then as a system of four Volterra integral equations. The latter permits a proof of existence and uniqueness of the solution of the original problem. Small and large time asymptotic solutions are derived and, from the first characterization, a regular perturbation solution is obtained. Numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin.

Numerical solutions to the equations describing Ericksen–Leslie dynamic theory for 2D nematic liquid crystal flows subject to a magnetic field are obtained. The governing equations are solved by a finite difference technique based on the GENSMAC methodology. The resulting numerical technique was verified by comparing numerical solutions for 2D-channel flow by means of mesh refinement. To demonstrate the capabilities of this method, the flow of a nematic liquid crystal in a planar 1:4 expansion was simulated. Calculations were performed for various Ericksen and Reynolds numbers. The results showed that an increase in the Ericksen number caused the appearance of lip and corner vortices.

In various attempts to relate the behaviour of highly-elastic liquids in complex flows to their rheometrical behaviour, obvious candidates for study have been the variation of shear viscosity with shear rate, the two normal stress differences N-1 and N-2, especially N-1, the extensional viscosity, and the dynamic moduli G' and G ''. In this paper, we shall confine attention to 'constant-viscosity' Boger fluids, and, accordingly, we shall limit attention to N-1, eta(E), G' and G ''. We shall concentrate on the "splashing" problem (particularly that which arises when a liquid drop falls onto the free surface of the same liquid). Modern numerical techniques are employed to provide the theoretical predictions. We show that high eta(E) can certainly reduce the height of the so-called Worthington jet, thus confirming earlier suggestions, but other rheometrical influences (steady and transient) can also have a role to play and the overall picture may not be as clear as it was once envisaged. We argue that this is due in the main to the fact that splashing is a manifestly unsteady flow. To confirm this proposition, we obtain numerical simulations for the linear Jeffreys model. (C) 2010 Elsevier B.V. All rights reserved.

In the analysis of stability of a variant of the Crank–Nicolson (C–N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C–N method and prove that their eigenvalues are inside [−1,1] for all values of m (the order of the matrix) and all values of a positive parameter σ, the stability parameter. As the order of the matrix is general, and the parameter σ lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.

This article deals with the study of the development and application of the high-order upwind ADBQUICKEST scheme, an adaptative bounded version of the QUICKEST for unsteady problems (Commun. Numer. Meth. Engng 2007; 23:419-445), employing both linear and nonlinear convection term discretization. This scheme is applicable to a wide range of computational fluid dynamics problems, where transport phenomena are of special importance. In particular, the performance of the scheme is assessed through an extensive numerical simulation study of advection-diffusion problems. The scheme, implemented in the context of finite difference methodology, combines a good approximation of shocks (or discontinuities) with a good approximation of the smooth parts of the solutions. In order to assess the performance of the scheme, seven problems are solved, namely (a) advection of scalars; (b) non-linear viscous Burgers equation; (c) Euler equations of gas dynamics; (d) Newtonian flow in a channel; (e) axisymmetric Newtonian jet flow; (f) axisymmetric non-Newtonian (generalized Newtonian) flow in a pipe; and (g) collapse of a fluid column. The numerical experiments clearly show that the scheme provides more consistent solutions than those found in the literature. From the study, the flexibility and robustness of the ADBQUICKEST scheme is confirmed by demonstrating its capability to solve a variety of linear and nonlinear problems with and without discontinuous solutions.

In this article recent advances in the Marker and Cell (MAC) method will be reviewed. The MAC technique dates back to the early 1960s at the Los Alamos Laboratories and this article starts with a historical review, and then a brief discussion of related techniques. Improvements since the early days of MAC (and the Simplified MAC - SMAC) include automatic time-stepping, the use of the conjugate gradient method to solve the Poisson equation for the corrected velocity potential, greater efficiency through stripping out the virtual particles (markers) other than those near the free surface, and more accurate approximations of the free surface boundary conditions, the addition of bounded high accuracy upwinding for the convected terms (thereby being able to solve higher Reynolds number flows), and a (dynamics) flow visualization facility. More recently, effective techniques for surface and interfacial flows and, in particular, for accurately tracking the associated surface(s)/interface(s) including moving contact angles have been developed. This article will concentrate principally on a three-dimensional version of the SMAC method. It will eschew both code verification and model validation; instead it will emphasize the applications that the MAC method can solve, from multiphase flows to rheology.

This paper describes the development of an implicit finite difference method for solving transient three-dimensional incompressible free surface flows. To reduce the CPU time of explicit low-Reynolds number calculations, we have combined a projection method with an implicit technique for treating the pressure on the free surface. The projection method is employed to uncouple the velocity and the pressure fields, allowing each variable to be solved separately. We employ the normal stress condition on the free surface to derive an implicit technique for calculating the pressure at the free surface. Numerical results demonstrate that this modification is essential for the construction of methods that are more stable than those provided by discretizing the free surface explicitly. In addition, we show that the proposed method can be applied to viscoelastic fluids. Numerical results include the simulation of jet buckling and extrudate swell for Reynolds numbers in the range [0.01,0.5].

This paper considers the stability of explicit, implicit and Crank-Nicolson schemes for the one-dimensional heat equation on a staggered grid. Furthermore, we consider the cases when both explicit and implicit approximations of the boundary conditions are employed. Why we choose to do this is clearly motivated and arises from solving fluid flow equations with free surfaces when the Reynolds number can be very small, in at least parts of the spatial domain. A comprehensive stability analysis is supplied: a novel result is the precise stability restriction on the Crank-Nicolson method when the boundary conditions are approximated explicitly, that is, at t=nt rather than t=(n+1)t. The two-dimensional Navier-Stokes equations were then solved by a marker and cell approach for two simple problems that had analytic solutions. It was found that the stability results provided in this paper were qualitatively very similar, thereby providing insight as to why a Crank-Nicolson approximation of the momentum equations is only conditionally stable.

This chapter provides a brief overview of the philosophy of mathematical modelling before providing three examples in the area of medical science. The first concerns drug-eluting stents; the second deals with blood flow in arteries; and the third concerns a capillary-fill device whose best known application is as part of a pregnancy testing kit. In each case the models are simplifications of the underlying physical and biological problems, yet they still provide useful insights. Mathematical modelling of drug release from drug-eluting stents, and the resulting arterial drug distributions is of real importance in the design and optimisation of such devices since the alternative (experimentation) is often prohibitively expensive and may raise ethical issues. Modelling flow in pipes has a long history dating back to the early 20th century; as our second illustration of modelling techniques we demonstrate how the mathematics behind this early research can be developed and applied to model blood flow in arteries. In our third example, we present a nonlinear mathematical model which, using transform theory and other techniques, facilitated the design of one of the most useful medical diagnostic tools in contemporary society-the pregnancy testing kit.

This paper is concerned with the numerical solutions of time dependent two-dimensional incompressible flows. By using the primitive variables of velocity and pressure, the Navier-Stokes and mass conservation equations are solved by a semi-implicit finite difference projection method. A new bounded higher order upwind convection scheme is employed to deal with the non-linear (advective) terms. The procedure is an adaptation of the GENSMAC (J. Comput. Phys. 1994; 110:171-186) methodology for calculating confined and free surface fluid flows at both low and high Reynolds numbers. The calculations were performed by using the 2D version of the Freeflow simulation system (J. Comp. Visual. Science 2000; 2:199-210). In order to demonstrate the capabilities of the numerical method, various test cases are presented. These are the fully developed flow in a channel, the flow over a backward facing step, the die-swell problem, the broken dam flow, and an impinging jet onto a flat plate. The numerical results compare favourably with the experimental data and the analytical solutions.

The present work is concerned with a semi-implicit modification of the GENSMAC method for solving the two-dimensional time-dependent incompressible Navier-Stokes equations in primitive variables formulation with a free surface. A projection method is employed to uncouple the velocity components and pressure, thus allowing the solution of each variable separately (a segregated approach). The viscous terms are treated by the implicit backward method in time and a centered second order method in space, and the nonlinear convection terms are explicitly approximated by the high order upwind variable-order nonoscillatory scheme method in space. The boundary conditions at the free surface couple the otherwise segregated velocity and pressure fields. The present work proposes a method that allows the segregated solution of free surface flow problems to be computed by semi-implicit schemes that preserve the stability conditions of the related coupled semi-implicit scheme. The numerical method is applied to both the simulation of free surface and to confined flows. The numerical results demonstrate that the present technique eliminates the parabolic stability restriction required by the original explicit GENSMAC method, and also found in segregated semi-implicit methods with time-lagged boundary conditions. For low Reynolds number flows, the method is robust and very efficient when compared to the original GENSMAC method.

This work presents a method for simulating axisymmetric and planar free-surface flows dominated by surface tension forces. The surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is evaluated by fitting a least squares curve to the free surface using the tracking markers in the cell and in its close neighbors. To avoid short wavelength perturbations on the free surface, a mass-conserving local four-point stencil filter is employed. This filter is an extension of the trapezoidal subgrid undulations removal (TSUR) method. The TSUR technique consists of modifying the positions of two ``internal'' markers of the four-point stencil in such a way that the surface length and the curvature are minimized, while still preserving the volume. The computation of the curvature is modified at cells adjacent to solid boundaries in order to apply contact angle boundary conditions. To identify neighboring cells efficiently, an implementation is effected through a dual representation of the cell data: in addition to a matrix representation, a list structure is also employed which permits the representation of specific groups of cells and associated data. This technique was implemented in the GENSMAC code, and it has been proved to be robust. The code is shown to produce accurate results when compared with exact solutions of selected fluid dynamical problems involving surface tension. Additionally, it is demonstrated that the method is applicable to complex free-surface flows, such as containers filling, splashing drops, and bursting bubbles.

The thermostat controller for an air-conditioning system is usually placed in a position at some distance from the unit and this can lead to large swings in temperature. This paper addresses this question by studying a paradigm - a one-dimensional heat conduction equation with and without heat loss, and where the flux of heat extracted or input by the unit is considered to be a function of the temperature at the other end. The essential results are that the system can be unstable and that this is exacerbated both by a more powerful air-conditioning unit and by more efficient insulation.

A method for simulating incompressible, imiscible, unsteady, Newtonian, multi-fluid flows with free surfaces is described. A sharp interface separates fluids of different density and viscosity. Surface and interfacial tensions are also considered and the required curvature is geometrically approximated at the fronts by a least squares quadratic fitting. To remove small undulations at the fronts, a mass-conserving filter is employed. The numerical method employed to solve the Navier-Stokes equations is based on the GENSMAC-3D front-tracking method. The velocity field is computed using a finite-difference scheme on an Eulerian grid. The free-surface and the interfaces are represented by an unstructured Lagrangian grid moving through an Eulerian grid. The method was validated by comparing the numerical results with analytical results for a number of simple problems. Complex numerical simulations show the capability and emphasize the robustness of this new method.

In this article recent advances in the MAC method will be reviewed. The MAC technique dates back to the early sixties at the Los Alamos Laboratories and this paper starts with a historical review, and then a summary of related techniques. Improvements since the early days of MAC (and the Simplified MAC-SMAC) include automatic time-stepping, the use of the conjugate gradient method to solve the Poisson equation for the corrected velocity potential, greater efficiency through stripping out all particles (markers) other than those near the free surface, more accurate approximations of the free surface boundary conditions, the addition of a bounded high accuracy upwinding for the convected terms (thereby being able to solve higher Reynolds number flows), and a (dynamic) flow visualization facility. This article will concentrate, in the main, on a three-dimensional version of the SMAC method. It will show how to approximate curved boundaries by considering one configurational example in detail; the same will also be done for the free surface. The article will avoid validation, but rather focus on many of the examples and applications that the MAC method can solve from turbulent flows to rheology. It will conclude with some speculative comments on the future direction of the methodology.

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time-averaged Navier-Stokes equations is achieved by using the two-equation eddy-viscosity model: the high-Reynolds k- (standard) model, with a time scale proposed by Durbin; and a low-Reynolds number form of the standard k- model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non-linear terms, a second/third-order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high-Reynolds k- model yields favourable predictions both of the zero-pressure-gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low-Reynolds number form of the k- model are somewhat unsatisfactory.

A numerical method for solving three-dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier-Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user-friendly input data file is employed to allow almost any arbitrary three-dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented.

Two-dimensional first-kind Volterra integral equations (VIEs) are studied. The first-kind equations are reduced to second kind, and by obtaining an appropriate integral inequality, existence and uniqueness are demonstrated. The equivalent discrete integral inequality then permits convergence of discretization methods; and this is illustrated for the Euler method. Finally, a class of nonlinear telegraph equations is shown to be equivalent to (two-dimensional) Volterra integral equations, thereby providing existence and uniqueness results for this class of equations. Furthermore, the telegraph equation may be solved by the numerical method for two-dimensional VIEs, and a simple numerical example is given.

This paper describes an extension of the GENSMAC code for solving two-dimen-sional free surface flows to axisymmetric flows. Like GENSMAC the technique is finite difference based and embodies, but considerably extends, the SMAC (simplified marker and cell) ideas. It incorporates adaptive time stepping and an accurate representation of the free surfaces while at the same time only uses surface particles to define the free surfaces, greatly increasing the computational speed; in addition, it employs a graphic interface with solid modeling techniques to provide enhanced three-dimensional visualization. Various simulations are undertaken to illustrate and validate typical flows. Both G. I. Taylor's viscous jet plunging into a fluid and a liquid drop splashing onto a fluid are simulated. Also, the important industrial application of container filling is illustrated. Finally, a comparison is made with the linear theory of standing waves and the code is validated by a numerical convergence study.

The electrohydrodynamic flow of a fluid in an “ion drag” configuration in a circular cylindrical conduit is governed by a non-linear second-order ordinary differential equation. The degree of non-linearity in this equation is determined by a nondimensional parameter α and the equation can be approximated by two different linear equations for very small or very large values of α respectively. Perturbation solutions of the fluid velocities for α ≪ 1 and α ≫ 1 are developed. A Gauss-Newton finite-difference solver combined with the continuation method and a Runge-Kutta shooting method are used to provide numerical results for the fluid velocity over a large range of values of α. Both numerical and analytical results are compared in order to establish the range of validity for the perturbation solutions. Die elektrohydrodynamische Stromung in einer „ion-drag”-Konfiguration in einem kreiszylindrischen Rohr wird durch eine nichtlineare gewohnliche Differentialgleichung 2. Ordnung beschrieben. Der Grad der Nichtlinearitat dieser Gleichung wird durch einen dimensionslosen Parameter a bestimmt, und die Gleichung kann durch zwei verschiedene lineare Gleichungen fur sehr kleine bzw. sehr grose Werte von α approximiert werden. Die Storungslosungen der Fluidgeschwindigkeiten fur α ≪ 1 und α ≫ 1 werden hergeleitet. Ein Gaus-Newton Finite-Differenz-Losungsverfahren, kombiniert mit der Fortsetzungsmethode und einem Runge-Kutta Shooting-Verfahren werden benutzt, um numerische Resultate fur die Stromungsgeschwindigkeit uber einen grosen α-Bereich herzuleiten. Die numerischen und analytischen Resultate werden verglichen, um den Gultigkeitsbereich der Storungslosungen festzustellen.

This paper is concerned with the study of the time-dependent water and air flow in a penstock caused by closing the head gate. Two approximate techniques are employed. The first approach uses simple global arguments based on mass and momentum conservation, and the time-dependent Bernoulli equation is used to derive evolution equations for the fluxes of water and air through the various components of the system. To complement this approach, the time-dependent Navier-Stokes equations were solved with a substantially reduced Reynolds number. Both approaches suggested that the pressure in the penstock remained above 1 2 atm, answering a question posed by Electricorp, New Zealand.

An alternating direction implicit (ADI) scheme is introduced which is capable of solving a general parabolic equation in two space dimensions with mixed derivative and convective terms. In the case of constant coefficients the scheme is shown to be unconditionally stable. The study was motivated by the investigation of flow in a dye laser cell (a device used for the amplification of a laser beam), a simple model for which involves laminar flow in a two-dimensional symmetric channel subject to impulsive heating. Numerical results are presented for this problem, and the qualitative behaviour of the temperature distribution within the channel for different Peclet numbers is discussed.

This paper is concerned with the modelling of the evolution of a chemical reaction within a small cell. Mathematically the problem consists of a heat equation with nonlinear boundary conditions. Through an integro-differential equation reformulation, an asymptotic result is derived, a perturbation solution is developed, and a modified product integration method is discussed. Finally, an alternative integral formulation is presented which acts as a check on the previous results and permits high accuracy numerical solutions.

A practical high resolution upwind differencing scheme for the numerical solution of convection-dominated transport problems is presented. The scheme is based on TVD and CBC stability criteria and is implemented in the context of the finite difference methodology. The performance of the scheme is investigated by solving the 1D/2D scalar advection equations, 1D inviscid Burgers’ equation, 1D scalar convection–diffusion equation, 1D/2D compressible Euler’s equations, and 2D incompressible Navier–Stokes equations. The numerical results displayed good agreement with other existing numerical and experimental data.

This paper presents an extension of the Enestrom–Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown ( K , L ) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994) [6] .

In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (t − s) −½.

A computer program for solving two-dimensional incompressible viscous fluid flow in general domains is described. This is based on the simplified Marker and cell technique, but it has a number of novel features. A user-supplied data file of coordinates prescribes the fluid domain which can be quite general and needs only to be connected. With a view to parallelisation the momentum equations are solved explicitly, but an automatic step-changing routine optimises the stability restriction. A conjugate gradient solver is used to invert the discrete Poisson equation. An accurate approximation to the stress conditions on the free surface is adopted. The code is written in structured FORTRAN with features from FORTRAN 90. The efficacy of the code is illustrated by applying it to some industrial problems.

This work presents a finite difference technique for simulating three-dimensional free surface flows governed by the Upper-Convected Maxwell (UCM) constitutive equation. A Marker-and-Cell approach is employed to represent the fluid free surface and formulations for calculating the non-Newtonian stress tensor on solid boundaries are developed. The complete free surface stress conditions are employed. The momentum equation is solved by an implicit technique while the UCM constitutive equation is integrated by the explicit Euler method. The resulting equations are solved by the finite difference method on a 3D-staggered grid. By using an exact solution for fully developed flow inside a pipe, validation and convergence results are provided. Numerical results include the simulation of the transient extrudate swell and the comparison between jet buckling of UCM and Newtonian fluids.

This study is primarily concerned with the presentation of a review of a collection (which could be regarded as a "test set") of linear and nonlinear singular integro-differential equations with Cauchy kernels, all of which arise from practical applications in Applied Mathematics and Mathematical Physics. The main objective of this review is to provide numerical analysts and researchers interested in algorithm development with model problems of genuine scientific interest on which to test their algorithms. Brief details of the methodology of derivation of the equations are provided and, where possible, existence, uniqueness and asymptotic results are discussed. References are also given to other studies that have dealt with similar problems. The importance of carrying out the necessary mathematical analysis is emphasized for one class of problems where it is shown that the solution abruptly ceases to exist as a parameter is varied. It is further shown that developing asymptotic estimates for the behavior of the solutions is very often a crucial component in the design of effective numerical methods. The importance of regularization is discussed for a class of problems, specific conclusions are drawn and recommendations are discussed. An appendix contains further related problems that may be used for further comparison purposes.

This article is concerned with the detection of environmental ageing in adhesively bonded structures used in the aircraft industry. Using a transmission line approach a forward model for the reflection coefficients is constructed and is shown to have an analytic solution in the case of constant permeability and permittivity. The inverse problem is analysed to determine necessary conditions for a unique recovery. The main thrust of this article then involves modelling the connector and then experimental rigs are built for the case of the air-filled line to enable the connector parameters to be identified and the inverse solver to be tested. Some results are also displayed for the dielectric-filled line.

In the present work we describe a method which allows the incorporation of surface tension into the GENSMAC2D code. This is achieved on two scales. First on the scale of a cell, the surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is estimated by fitting a least square circle to the free surface using the tracking particles in the cell and in its close neighbors. On a sub-cell scale, short wavelength perturbations are filtered out using a local 4-point stencil which is mass conservative. An efficient implementation is obtained through a dual representation of the cell data, using both a matrix representation, for ease at identifying neighbouring cells, and also a tree data structure, which permits the representation of specific groups of cells with additional information pertaining to that group. The resulting code is shown to be robust, and to produce accurate results when compared with exact solutions of selected fluid dynamic problems involving surface tension.

In this paper we consider numerical solutions to a kinetic equation for the dispersion of small particles in a turbulent flow. The solution represents the probability density that a particle has a certain velocity and position at a given time. These solutions are based on a mixed finite-difference spectral method. Computational results are presented.

A convergence theorem is given showing that zero-stable advanced linear multistep methods with orderp consistency have orderp convergence.

The construction and convergence of high-order product integration methods for the second-kind Abel equation are discussed and the results of De Hoog and Weiss are generalised. Backward difference methods are introduced, and numerical results are presented which verify the theoretical rates of convergence.

This paper is concerned with Volterra integral equations of the first kind whose kernel,k(t,s), is identically zero whent=s. The concepts of zero-stability and weak zero-stability are introduced and convergence results under the assumption that the truncation error has an asymptotic expansion with a certain number of terms are presented. Simple numerical examples verifying these rates of convergence are given.

A method is described based on auniform mesh for the singular two-point boundary value problem:y″+(α/x)y′+f(x, y)=0, 0

L'etude est basee sur l'equation de mouvement lent d'une particule spherique (Basset, Boussinesq et Oseen) pour un fluide au repos, etendue par Tchen au fluide en mouvement