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This paper is concerned with the numerical solution of high-dimensional Fokker-\ud Planck equations related to multi-dimensional diffusion with polynomial coefficients\ud or Pearson diffusions. Classification of multi-dimensional Pearson diffusion follows\ud from the classification of one-dimensional Pearson diffusion. There are six important\ud classes of Pearson diffusion - three of them possess an infinite system of moments\ud (Gaussian, Gamma, Beta) while the other three possess a finite number of moments\ud (inverted Gamma, Student and Fisher-Snedecor). Numerical approximations to the\ud solution of the Fokker-Planck equation are generated using the spectral method.\ud The use of an adaptive reduced basis technique facilitates a significant reduction in\ud the number of degrees of freedom required in the approximation through the determination\ud of an optimal basis using the singular value decomposition (SVD). The\ud basis functions are constructed dynamically so that the numerical approximation\ud is optimal in the current finite-dimensional subspace of the solution space. This is\ud achieved through basis enrichment and projection stages. Numerical results with different\ud boundary conditions are presented to demonstrate the accuracy and efficiency\ud of the numerical scheme.

Proper generalised decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two different reformulations of the problem. We show that these least-squares PGD provide a much stabler algorithm than an equivalent Galerkin PGD and provide proofs of convergence of the algorithms.

This paper is concerned with a comparison of the performance and efficiency of mixed finite element methods for solving single phase fluid flow in porous media. Particular attention is given to the accurate determination of the groundwater fluxes. The linear systems generated by the mixed finite element method (MFEM) are indefinite. Symmetric positive definite linear systems are generated through the introduction of Lagrange multipliers giving rise to the mixed hybrid finite element method (MHFEM). The convergence behaviour of the numerical approximations is investigated over a range of conductivity coefficients from heterogeneous, isotropic and diagonal to discontinuous, anisotropic and full, on both triangular and quadrilateral, structured and distorted meshes. The robustness and efficiency of various preconditioned solvers is investigated in terms of optimality with respect to both mesh size and conductivity coefficient.

This paper is concerned with the development of efficient iterative methods for solving the linear system of equations arising from stochastic FEMs for single-phase fluid flow in porous media. It is assumed that the conductivity coefficient varies randomly in space according to some given correlation function and is approximated using a truncated Karhunen–Loeve expansion. Distinct discretizations of the deterministic and stochastic spaces are required for implementations of the stochastic FEM. In this paper, the deterministic space is discretized using classical finite elements and the stochastic space using a polynomial chaos expansion. The highly structured linear systems which result from this discretization mean that Krylov subspace iterative solvers are extremely effective. The performance of a range of preconditioned iterative methods is investigated and evaluated in terms of robustness with respect to mesh size and variability of the conductivity coefficient. An efficient symmetric block Gauss–Seidel preconditioner is proposed for problems in which the conductivity coefficient has a large standard deviation.The companion paper, herein, referred to as Part 2, considers the situation in which Gaussian random fields are transformed into lognormal ones by projecting the truncated Karhunen–Loeve expansion onto a polynomial chaos basis. This results in a stochastic nonlinear problem because the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables.

The simplest and most efficient lattice Boltzmann model that is able to recover the Navier-Stokes equations is based on a single-parameter scattering matrix, where the parameter is the first nonzero eigenvalue of the collision matrix. This simple model, based on a single relaxation time, has many shortcomings. Among these is the lack of freedom to extend the model to complex fluids whose stress tensors are characterized by more complicated constitutive relations. The lattice Boltzmann methodology may be generalized by considering the full collision matrix and tuning the matrix elements to obtain the desired macroscopic properties. The generalized hydrodynamics of a generalized lattice Boltzmann equation (LBE) was studied by Lallemand and Luo [Phys. Rev. E 61, 6546 (2000)] by solving the dispersion equation of the linearized LBE. In this paper, an alternative approach to solving the dispersion equation based on a formal perturbation analysis is described. The methodology outlined is systematic, can be readily applied to other lattices, and does not require the reciprocals of the relaxation times to be small.

Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.

The start-up of plane Couette flow of a finitely extensible nonlinear elastic (FENE) fluid is considered. A numerical method for solving this problem based on a decoupled micro{macro approach is described. The polymeric stress is determined in the microscopic part of the calculation by computing the solution of a Fokker-Planck equation for the configuration probability density function. The solution of this equation often suffers from the so-called problem of "curse of dimensionality." An efficient solution procedure is described based on the use of an adaptive reduced basis function technique in conjunction with high-order spectral approximations. A substantial reduction in the number of degrees of freedom to achieve a required level of accuracy is obtained compared with standard tensor product representation of the basis. Some sample numerical results are presented to highlight properties of the model and the numerical scheme.

The numerical solution of the one-dimensional Fokker-Planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer dynamics is presented. The finitely extensible non-linear elastic (FENE) model is considered and the spectral element discretisation is applied using an adaptive reduced basis technique. This technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection. The reduced basis method is extended to the high-dimensional Fokker-Planck equation, using the d-dumbbell FENE model, by consideration of a high-dimensional singular value decomposition. Some numerical results are presented to demonstrate the efficiency of the numerical scheme.

This paper presents a modification to the numerical treatment of integral constitutive equations initiated by Peters et al. [E.A.J.F. Peters, M.A. Hulsen, B.H.A.A. van den Brule, Instationary Eulerian viscoelastic flow simulations using time separable Rivlin–Sawyers constitutive equations. J. Non-Newtonian Fluid Mech., 89 (2000) 209–228] that enables computational stability to be achieved for higher Weissenberg number for steady flows. Increased efficiency is also achieved so that computational times are of the same order (2 or 3 times) as equivalent differential models and of similar accuracy. The memory cost for storage of the deformation fields is approximately 300 times that of the stress tensor for each grid point to yield approximations that lie within 0.01% of predictions of equivalent differential models. The good agreement between equivalent integral and differential models found here suggests that more general integral models with no differential equivalent may be used reliably for steady flow simulations. A modification of the deformation fields method that avoids the introduction of a cut-off time and which is applicable to transient flows is also presented.

The particle deficiency problem in the presence of a rigid wall for smoothed particle hydrodynamics (SPH) is considered. The problem arises from insufficient information being available to perform accurate interpolation of data at particles located nearer to the boundary than the support of the interpolation kernel. The standard method for overcoming this problem is based on the introduction of image particles to populate the deficient regions and the use of linear extrapolation to determine the velocity of these image particles from that of fluid particles. A consistent treatment of boundary conditions, utilising the momentum equation to obtain approximations to the velocity of image particles, is described. The method ensures second order approximation of the boundary conditions. It is validated for Poiseuille and Couette flow, for which analytical series solutions exist and shows second order convergence under certain conditions.

The problem of droplet deformation and break-up is considered. A hybrid Eulerian-Lagrangian method is used in which the velocity and pressure are discretized on a fixed mesh and Lagrangian particles are used to implicitly track the interface between the two phases. The Navier-Stokes equations are solved using an approximate Godunov projection method, collectively called the Godunov marker-particle projection scheme. The results show good qualitative agreement with previous research as well as demonstrating the efficacy of the method.

New spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared.

This paper considers the spectral element approximation of the Stokes problem and the conditioning of the resulting discrete problem. The well-posedness of the variational formulation of the Stokes problem and, in particular, the uniqueness of the pressure has been demonstrated when the subspace of square-integrable functions having vanishing mean is chosen for the pressure space. In the discrete setting a conforming subspace is chosen for the discrete pressure space and the analysis of the discrete problem is based on this choice. However, this space is not used in practical implementations of finite or spectral element methods for the Stokes problem. In this paper it is shown how the zero volume condition on the pressure may be accommodated within the trial space by modifying the continuity equation in a consistent manner. The dependence of the condition number of the Uzawa operator and the accuracy of the spectral element approximation on the weighting factor is investigated.

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.

In this paper accurate and stable finite volume schemes for solving viscoelastic flow problems are presented. Two contrasting finite volume schemes are described: a hybrid cell-vertex scheme and a pure cell-centred counterpart. Both schemes employ a time-splitting algorithm to evolve the solution through time towards steady state. In the case of the hybrid scheme, a semi-implicit formulation is employed in the momentum equation, based on the Taylor-Galerkin approach with a pressure-correction step to enforce incompressibility. The basis of the pure finite volume approach is a backward Euler scheme with a semi-Lagrangian step to treat the convection terms in the momentum and constitutive equations. Two distinct finite volume schemes are presented for solving the systems of partial differential equations describing the flow of viscoelastic fluids. The schemes are constructed to be second-order accurate in space. The issue of stability is also addressed with respect to the treatment of convection. Numerical examples are presented illustrating the performance of these schemes on some steady and transient problems that possess analytical solutions.

The paper is concerned with the numerical prediction of the flow of polymer melts using a pom-pom model [J. Rheol. 42 (1998) 81]. The pom-pom model is a coarse-grained molecular model that was developed for describing branched polymers. A description of the configuration distribution is given in terms of the orientation and stretch of the pom-pom molecule. The variant of the pom-pom model used in this paper is the extended pom-pom model [J. Rheol. 45 (4) (2001) 823]. The model has been rewritten to make it suitable for implementation in an existing spectral element algorithm, which solves the constitutive equation together with the momentum and continuity equations The temporal discretization of the convective terms in the momentum equation and the constitutive equation is performed using a second-order time-splitting technique. Spatial discretization is performed using spectral element techniques. The system of discretized equations is solved using a preconditioned conjugate gradient method, and details of the preconditioners are given. Results are presented for the start-up of Poiseuille flow in a planar channel, and flow past a confined cylinder. The influence of the model parameters on the solution is described.

The flow of a viscoelastic fluid through an undulating tube is considered. The fluid is modelled using the UCM constitutive equation. The governing set of equations is solved using a time-splitting technique. This is based on separate treatments of the convection and generalised Stokes operators. The spatial discretisation is based on a spectral discretisation in which the radial basis functions satisfy the conditions along the axis of symmetry of the tube explicitly. Compatible approximation spaces are chosen for velocity, pressure and extra-stress. The convection problem is solved in a type-sensitive manner using a high-order Runge–Kutta method. The weak formulation of the generalised Stokes problem is discretised and solved using a nested conjugate gradient technique. The effect of elasticity on flow resistance is investigated and comparisons made with other results in the literature.

This paper is devoted to the development of efficient iterative methods for solving the systems of algebraic equations arising from a spectral element discretization of the equations governing the flow of an Oldroyd B fluid. The governing equations are written in terms of velocity, pressure and extra-stress, giving rise to the so-called three-field formulation. The convection terms are treated using an operator-integration-factor splitting method. The remaining terms in the system of equations constitute a generalized Stokes problem. This problem is formulated in terms of an Uzawa operator applied to the pressure. Efficient preconditioners are developed for inverting this operator that are independent of the numerical and physical parameters of the problem.

The approximation of the Stokes problem in axisymmetric geometries using the spectral element method is considered. The presence of the volume element r dr dz in the weak formulation of the problem is shown to be a potential source of difficulty. The discrete equations associated with nodes on the axis of symmetry can lead to a degeneracy in the global system of equations. This difficulty is resolved by incorporating the factor r into the weight function for spectral elements adjacent to the axis of symmetry and using appropriate basis functions in these elements in the radial direction. Properties of the Jacobi polynomials are used to construct the elements of the modified method. Numerical results are presented demonstrating some of the features of the proposed approach.

The numerical approximation of the mixed velocity–pressure–stress formulation of the Stokes problem using spectral methods is considered. In addition to the compatibility condition between the discrete velocity and pressure spaces a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well-posed problem. The theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational. Proper matching conditions between spectral elements are addressed leading to a discontinuous extra-stress approximation.

The two-dimensional flow of a Newtonian fluid past a cylinder is investigated. The governing equations are discretized using a second-order temporal discretization based on the method of characteristics and a spectral element spatial discretization. The resulting discrete systems are symmetric and (semi-) positive definite and are solved using the conjugate gradient method. The performance of the algorithm is examined. The dependence of the drag on the cylinder as a function of the Reynolds number is investigated.

The numerical approximation of the mixed velocity-pressure-stress formulation of the Stokes problem using spectral methods is considered. In addition to the compatibility condition between the discrete velocity and pressure spaces, a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well-posed problem. The theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational. The discrete problem is analyzed and error estimates are derived. A comparison between the mixed approach and the standard velocity-pressure formulation is made in terms of the condition number of the resulting systems.

A muli-domain spectral collocation method is developed for the approximation of the Stokes problem in two dimensions. Compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well-posed. The collocation scheme possesses the property that the continuity equation is satisfied at all the collocation points. In addition, for problems defined in domains which are rectangularly decomposable, the scheme yields velocity approximations that are globally divergence-free.

The steady flow of a viscoelastic fluid past a sphere in a cylindrical tube is considered. A spectral element method is used to solve the system of coupled non-linear partial differential equations governing the flow. The spectral element method combines the flexibility of the traditional finite element method with the accuracy of spectral methods. A time-splitting algorithm is used to determine the solution to the steady problem. Results are presented for the Oldroyd B model. These show excellent agreement with the literature. The results converge with mesh refinement. A limiting Deborah number of approximately 0⋅6 is found, irrespective of the spatial resolution.

In this paper we present a new high-order method for the approximation of the Navier–Stokes equations in rectangularly decomposable domains. The governing equations are written in terms of a stream function. The resulting nonlinear fourth-order equation for the stream function is linearized using Newton’s method. An equivalent variational formulation of the linearized problem is derived from which a collocation scheme is constructed. Pseudospectral approximations which are $C^1 $ continuous at the interfaces between subdomains are used to approximate the solution. The approximations are shown to be $C^3 $ continuous at the interfaces asymptotically as the order of the polynomial approximation within each subdomain is increased. Numerical results are presented for a Stokes problem which show that the approximations are exponentially convergent. The method is used to solve the Navier–Stokes equations in an L-shaped domain for a range of Reynolds numbers. Stream function contours are displayed showing the ma...

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multi-domain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C^1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C^3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multi-domain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a non-rectangular domain.

The governing equations for Stokes flow are formulated in terms of a stream function and Airy stress function. This formulation ensures that mass and momentum are conserved identically. In terms of these new variables, the equations of motion are written as a second-order elliptic system. These equations are embedded in biharmonic equations and the boundary conditions appropriate for this higher-order system are determined using a least-squares process. This technique is applied to the planar stick-slip problem. A numerical solution to the problem is obtained using a spectral domain decomposition method. An algebraic mapping is used to treat the flow domain without truncation. The coefficients in a singular expansion of the stream function about the stick-slip singularity are computed using a post-processing technique.

Algorithms for the transient simulation of axisymmetric non-Newtonian flows using spectral methods are described for the primitive variable formulation of the governing equations. A modification of the method of Ku et al. (Comput. Fluids15, pp. 195–214 (1987)) and the construction of an optimal approximation space for the pressure ensures that the velocity field is globally divergence free at the end of each time step. The application of these techniques to some fundamental non-Newtonian flow problems is described.

A new pseudospectral method for simulating transient viscoelastic flows is presented. The governing equations are a system of seven first-order equations of mixed type. The essential features of the method are (i) all seven independent flow variables are represented on a common Chebyshev-Gauss-Lobatto grid; (ii) the pressure is treated in such a way as to give a globally divergence-free velocity field, i.e., the divergence of the velocity field vanishes globally within the region, and (iii) different time scales pertaining within the hyperbolic constitutive equations are treated using the splitting technique of LeVeque and Yee originally proposed in a finite-difference context. The method is applied to transient axisymmetric flow of an Oldroyd B fluid in a channel formulated in two ways: (I) as an initial boundary-value problem, and (II) as a body-force problem. © 1993 John Wiley & Sons, Inc.

Algorithms for the transient and steady state simulation of incompressible Newtonian and non-Newtonian flows are described for the primitive variable formulation of the governing equations. Spectral approximations are used for the spatial discretization. Attention is given to the satisfaction of the incompressibility constraint and the determination of the pressure. Spurious pressure modes are removed by means of a singular value decomposition. The corresponding velocity field is divergence free at all the collocation points. Numerical results are presented for Newtonian flow in a regularized driven square cavity and for non-Newtonian flow in a planar channel and in a journal bearing for a realistic range of material parameters.

When standard spectral methods are applied to the biharmonic equation the condition number of the resulting algebraic system of equations is very large and grows as 0(N 8) where N is the degree of the polynomial expansion in each coordinate direction. When direct methods are used to solve these systems we fail to achieve machine precision for the solution due to rounding errors. An improved method which reduces the condition number to 0(N 4) is described. Numerical results are presented demonstrating the exponential convergence of the approximation.

The flow past a sphere in an infinite expanse of viscoelastic fluid is simulated numerically using a pseudospectral technique. The problem is formulated using primitive variables. A time-splitting scheme is used that ensures that the velocity field is divergence-free at the end of each time step. The flow domain is treated without truncation using rational Chebyshev polynomials as basis functions in the radial direction. The drag on the sphere is computed for a range of values of the non-dimensional elasticity number We . A quantitative comparison of the drag coefficient with that obtained by a perturbation analysis is given. As We is initially increased, a reduction in the drag from its Newtonian value is observed. For larger values of We , drag enhancement is detected. In addition, for small values of We the streamlines are shifted slightly downstream in agreement with experimental observations. The numerical algorithm is described in detail and the numerical results compared qualitatively with experiments.

A review of preconditioned iterative techniques for elliptic problems on decomposed domains is given. Domain decomposition methods based on overlapping (Schwarz alternating method) and nonoverlapping (Schur complement method) subdomains are described for the model Poisson problem. An additive variant of the Schwarz method which is more suitable to implementation on parallel computers than the traditional multiplicative Schwarz method is presented. Some preconditioners for the Schur domain decomposition method which improve the rate of convergence of the underlying conjugate gradient method are suggested. Some remarks about nonsymmetric problems are made.

A spectral domain decomposition method for solving the planar stick-slip problem for an Oldroyd-B fluid is presented. The flow domain is decomposed into two semi-infinite rectangular subdomains. A comparison of the treatment of these subdomains by domain truncation and algebraic mapping is given. A fixed point method is used to linearize the constitutive equation. The linearized systems are solved in a manner sympathetic towards the structure of the coefficient matrices at each iterative step. The algorithm employed in solving the stick-slip problem is shown to be efficient in its demand upon array storage and CPU time. The numerical results demonstrate the necessity of adequate entry and exit lengths for the non-Newtonian problem and the adverse effect that the singularity has upon the range of Weissenberg numbers over which converged solutions are possible. Results generated from a regularized stick-slip geometry where the boundary singularity is smoothed out provide evidence that it is possible to extend this range by solving a slightly modified problem in which the effect of the singularity is weakened.

The application of the finite difference method to discretize Poisson's equation in an annular region produces an algebraic system which reflects the periodic nature of the problem. The two natural orderings of the unknowns yield coefficient matrices which are either block cyclic tridiagonal or block tridiagonal with diagonal blocks which are cyclic tridiagonal. The use of methods based on incomplete LU decompositions for solving these systems are examined. An explicit treatment of the periodicity in the incomplete LU factorization of the coefficient matrix is avoided by using the method of deletion and complementation. The method, when used as a preconditioner for the minimal residual method, is found to be competitive with other methods based on factorization ideas and to be more robust.

This study investigates the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We extend our earlier work for Poiseuille flow in a planar channel and the single equation form of the extended pom-pom (SXPP) model [M. Aboubacar, J.P. Aguayo, P.M. Phillips, T.N. Phillips, H.R. Tamaddon-Jahromi, B.A. Snigerev, M.F. Webster, Modelling pom-pom type models with high-order finite volume schemes, J. Non-Newtonian Fluid Mech. 126 (2005) 207-220], to determine steady-state solutions for planar 4:1 sharp contraction flows. The numerical techniques employed are time-stepping algorithms: one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum-continuity equations by a fractional-staged Taylor-Galerkin pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. A comparison of the two finite volume approaches is presented, concentrating upon the new features posed by the pom-pom class of models in this context of non-smooth flows. Here, the dominant feature of larger shear and extension in the entry zone influences both stress and stretch, so that larger stretch develops around the re-entrant corner zone as Weissenberg number increases, whilst correspondingly stress levels decline.

A new finite volume method for solving the incompressible Navier--Stokes equations is presented. The main features of this method are the location of the velocity components and pressure on different staggered grids and a semi-Lagrangian method for the treatment of convection. An interpolation procedure based on area-weighting is used for the convection part of the computation. The method is applied to flow through a constricted channel, and results are obtained for Reynolds numbers, based on half the flow rate, up to 1000. The behavior of the vortex in the salient corner is investigated qualitatively and quantitatively, and excellent agreement is found with the numerical results of Dennis and Smith [ Proc. Roy. Soc. London A, 372 (1980), pp. 393--414] and the asymptotic theory of Smith [J. Fluid Mech., 90 (1979), pp. 725--754].

La region d'ecoulement est decomposee en un certain nombre de sous regions rectangulaires plus simples

A spectral element method is described for solving Stokes flow in rectangularly decomposable domains. The flow region is divided into a number of rectangular subregions, some of which may be semi-infinite. The solution to the governing biharmonic equation for the stream function is represented by an expansion of Chebyshev polynomials in each subregion. The coefficients in these expansions are determined by collocating the differential equation and boundary conditions, and imposing $C^3 $ continuity across subregion interfaces. The spectral element method produces systems of equations that are block tridiagonal. Efficient direct methods based on capacitance matrix ideas are proposed that take advantage of the structure of the spectral element matrix. The use of these techniques produces savings in storage and time.

Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem.

The systems of algebraic equations which arise from spectral discretizations of elliptic equations are full and direct solutions of them are rarely feasible. Iterative methods are an attractive alternative because Fourier transform techniques enable the discrete matrix-vector products to be computed with nearly the same efficiency as is possible for corresponding but sparse finite difference discretizations. For realistic Dirichlet problems preconditioning is essential for acceptable convergence rates. A brief description of Chebyshev spectral approximations and spectral multigrid methods for elliptic problems is given. A survey of preconditioners for Dirichlet problems based on second-order finite difference methods is made. New preconditioning techniques based on higher order finite differences and on the spectral matrix itself are presented. The preconditioners are analyzed in terms of their spectra and numerical examples are presented.

A spectral element method is described which enables Poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi-infinite rectangular regions. Two choices of trial functions are considered, namely the eigenfunctions of the differential operator and Chebyshev polynomials. The coefficients in the series expansions are obtained by imposing weak C1 matching conditions across element interfaces. Singularities at re-entrant corners are treated by a post-processing technique which makes use of the known asymptotic behaviour of the solution at the singular point. Accurate approximations are obtained with few degrees of freedom.

A compact difference scheme is described for treating the first order system of PDE's which describe the equilibrium equations of an elastic body. An algebraic simplification enables the solution to be obtained by standard direct or iterative techniques.

Second order equations in terms of auxiliary variables similar to potential and stream functions are obtained by applying a weighted least squares formulation to a first order system. The additional boundary conditions which are necessary to solve the higher order equations are determined and numerical results are presented for the Cauchy-Riemann equations.

The performance of the alternating direction implicit (ADI) method within the multigrid context is examined. Local mode analysis is used to predict the smoothing properties of ADI for the Laplace problem and the biharmonic equation. The smoothing rate of ADI is shown to compare favourably with that of the Gauss-Seidel method. As a result, fewer iterations are required when ADI is used as the relaxation process in the multigrid method.

Techniques are considered for improving the numerical approximation to a problem which possesses corner singularities. The particular problem considered here is taken from solid state electronics. Relined numerical solutions are obtained using two techniques. The first employs a non-uniform grid with a smaller spacing of the mesh points in the neighbourhood of the singularities. The second technique involves a transformation of the problem to one which is free from singularities. The dynamic ADI method, which needs no a priori relaxation parameters to accelerate convergence, is used to solve the discretized problem in both cases.

Spectral techniques for solving problems in non-Newtonian fluid mechanics are introduced. Following the work of Coleman (J. Non-Newtonian Fluid Mech.; 15, 227–238 [1984]), the governing equations for the creeping flow of a co-rotational Maxwell fluid are written in terms of the Airy stress function and a stream function. This ensures that the continuity and momentum equations are automatically satisfied. The choice of trial functions for solving a one-dimensional model problem using spectral methods is discussed. Methods for treating unbounded domains and accurately representing reentrant boundary singularities within the spectral context are also considered.

On etablit une relation entre les coefficients de Legendre d'une derivee generale d'une fonction indefiniment derivable et les coefficients de Legendre de la fonction elle-meme

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics. In recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations. Analytical solutions have been derived for some of the Pearson diffusions, including the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross and Jacobi processes. However, analytical investigations and computations for diffusions with so-called heavy-tailed ergodic distributions are more difficult to perform. The novelty of this research is the development of an accurate and efficient numerical method to solve the Fokker–Planck equations associated with Pearson diffusions with different boundary conditions. Comparisons between the numerical predictions and available time-dependent and equilibrium analytical solutions are made. The solution of the Fokker–Planck equation is approximated using a reduced basis spectral method. The advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving Fokker–Planck type equations.