Nonlinear high-wavenumber Bnard convection

Weakly nonlinear two-dimensional roll cells in Bnard convectionare examined in the limit as the wavenumber a of the roll cellsbecomes large. In this limit the second harmonic contributionsto the solution become negligible, and a flow develops wherethe fundamental vortex terms and the correction to the meanare determined simultaneously, rather than sequentially as inthe weakly nonlinear case. Extension of this structure to Rayleighnumbers O(a³) above the neutral curve is shown to be possible,with the resulting flow field having a form very similar tothat for strongly nonlinear vortices in a centripetally unstableflow. The flow in this strongly nonlinear regime consists ofa core region, and boundary layers of thickness O(a^–1)at the walls. The core region occupies most of the thicknessof the fluid layer and only mean terms and cos az terms playa role in determining the flow; in the boundary layer all harmonicsof the vortex motion are present. Numerical solutions of thewall layer equations are presented and it is also shown thatthe heat transfer across the layer is significantly greaterthan in the conduction state.     

Asymptotic Error Expansions for Numerical Solutions of Integral Equations

A general theorem dealing with asymptotic error expansions fornumerical solutions of linear operator equations is proved.This is applied to the Nystr?m, collocation, and Galerkin methodsfor second kind, Fredholm integral equations. For example, weshow that when piecewise polynomials of degree m–1 areused, the iterated Galerkin solution admits an error expansionin even powers of the step-size h, beginning with a term inh^2m.     

Spectral decompositions in nonlinear coupled diffusion

The solutions of a coupled, linear and nonlinear diffusion equationin a semi-infinite medium are derived using series methods.In addition, perturbation techniques allied to the spectraldecomposition of matrices are used to simplify the analysisand to find semianalytic solutions. The discussion is motivatedby the transmission of heat, moisture, and solute through thestrongly nonlinear medium of soil. Under boundary conditionsrepresenting the daily or seasonal fluctuations, it is shownusing spectral decomposition, despite the nonlinearities, howthe period of oscillation is preserved on passage through themedium. It is also shown how n³ partial differential equationsmay be solved for each of the n coupled variables to determineclosed forms for the first- and second-order perturbation effects.Examples of the solutions are given for the case of the coupledtransport of heat and moisture in soil.     

Locking-free DGFEM for elasticity problems in polygons

The h-version of the discontinuous Galerkin finite element method(h-DGFEM) for nearly incompressible linear elasticity problemsin polygons is analysed. It is proved that the scheme is robust(locking-free) with respect to volume locking, even in the absenceof H²-regularity of the solution. Furthermore, it is shown thatan appropriate choice of the finite element meshes leads torobust and optimal algebraic convergence rates of the DGFEMeven if the exact solutions do not belong to H².     

A Fast Method for the Solution of Fredholm Integral Equations

Methods described to date for the solution of linear Fredholmintegral equations have a computing time requirement of O(N³),where N is the number of expansion functions or discretizationpoints used. We describe here a Tchebychev expansion method,based on the FFT, which reduces this time to O(N² ln N), andreport some comparative timings obtained with it. We give alsoboth a priori and a posteriori error estimates which are cheapto compute, and which appear more reliable than those used previously.     

Stationary Critical Points of the Heat Flow in Spaces of Constant Curvature

The paper considers stationary critical points of the heat flowin sphere S^N and in hyperbolic space H^N, and proves severalresults corresponding to those in Euclidean space R^N which havebeen proved by Magnanini and Sakaguchi. To be precise, it isshown that a solution u of the heat equation has a stationarycritical point, if and only if u satisfies some balance lawwith respect to the point for any time. In Cauchy problems forthe heat equation, it is shown that the solution u has a stationarycritical point if and only if the initial data satisfies thebalance law with respect to the point. Furthermore, one point,say x₀, is fixed and initial-boundary value problems are consideredfor the heat equation on bounded domains containing x₀. It isshown that for any initial data satisfying the balance law withrespect to x₀ (or being centrosymmetric with respect to x₀)the corresponding solution always has x₀ as a stationary criticalpoint, if and only if the domain is a geodesic ball centredat x₀ (or is centrosymmetric with respect to x₀, respectively).     

Similarity Solutions of a Non-linear Diffusion Equation

In several physical contexts the equations for the dispersionof a buoyant contaminant can be approximated by the Erdogan-Chatwin(1967) equation {dot}c = {dot}_y{[D_o + ({dot}_yc)²D₂]{dot}_yc}. Here it is shown that in the limit of strong non-linearity (i.e.D_o = 0) there are similarity solutions for a concentration jumpand for a finite discharge. A stability analysis for the latterproblem involves a new family of orthogonal polynomials Y_n(z)where (1 – z⁴)Y – 6z³Y + n(n + 5)z² Y_n = 0 and the degree n is restricted to the values 0, 1, 4, 5, 8,9,.... A numerical solution of the Erdogan-Chatwin equationis given which describes the transition between the non-linearand linear (Gaussian) similarity solutions.     

Spurious solutions of numerical methods for initial value problems

It is well known that some numerical methods for initial valueproblems admit spurious limit sets. Here the existence and behaviourof spurious solutions of Runge-Kutta, linear multistep and predictor-correctormethods are studied in the limit as the step-size h0. In particular,it is shown that for ordinary differential equations definedby globally Lipschitz vector fields, spurious fixed points andperiod 2 solutions cannot exist for h arbitrarily small, whilstfor locally Lipschitz vector fields, spurious solutions mayexist for h arbitrarily small, but must become unbounded ash0. The existence of spurious solutions is also studied forvector fields merely assumed to be continuous, and an exampleis given, showing that in this case spurious solutions may remainbounded as h0. It is shown that if spurious fixed points orperiod 2 solutions of continuous problems exist for h arbitrarilysmall, then as h0 spurious solutions either converge to steadysolutions of the underlying differential equation or divergeto infinity. A necessary condition for the bifurcation spurioussolutions from h=0 is derived. To prove the above results forimplicit Runge-Kutta methods, an additional assumption on theiteration scheme used to solve the nonlinear equations definingthe method is needed; an example of a Runge-Kutta method whichgenerates a bounded spurious solution for a smooth problem withh arbitrarily small is given, showing that such an assumptionis necessary. It is also shown that an Adams-Bashforth/Adams-Moultonpredictor-corrector method in PC^m implementation can generatespurious fixed point solutions for any m.     

A Hitchin-Kobayashi Correspondence for Coherent Systems on Riemann Surfaces

A coherent system (E, V) consists of a holomorphic bundle plusa linear subspace of its space of holomorphic sections. Motivatedby the usual notion in geometric invariant theory, a notionof slope stability can be defined for such objects. In the paperit is shown that stability in this sense is equivalent to theexistence of solutions to a certain set of gauge theoretic equations.One of the equations is essentially the vortex equation (thatis, the Hermitian–Einstein equation with an additionalzeroth order term), and the other is an orthonormality conditionon a frame for the subspace VH⁰(E).     

Spatial decay in a cross-diffusion problem

In this paper, the authors investigate the decay of end effectsfor a cross-diffusion problem defined on a semi-infinite cylindricalregion. With homogeneous Dirichlet or Neumann conditions prescribedon the lateral surface of the cylinder, it is shown that forfixed finite time and under certain restrictions on the coefficients,solutions decay point-wise as the distance d from the finiteend of the cylinder tends to infinity at least of order e^–kd2.Under less restrictive conditions, it is shown that solutionsdecay in L₂ at least as fast as e^–kd. In both cases, kis a computable function of time.     

A Storage-Efficient Algorithm for Finding the Regularized Solution of a Large, Inconsistent System of Equations

We propose an algorithm which for any real number r, any k ?l matrix M and any k-vector y, finds the l-vector x which minimizes||x||² – r²||Mx – y||², i.e. it finds the "regularized"solution to the equation Mx = y. (|| || denotes the 2-norm.)The algorithm is iterative with the following properties: (i)in a single step it needs access to only one row of the matrixM, (ii) it needs to store only the present estimate of the solution(size l) and a "residual vector" of size k, (iii) in a singlestep it updates only one component of the residual vector. Becauseof these properties the algorithm has been found useful in "solving"very large inconsistent systems of equations. Convergence ofthe algorithm to the desired solution is proved and the rateof convergence of the algorithm is illustrated. The algorithmis considered here, not because it is believed to be generallysuperior to more commonly used iterative methods, but becausefor certain very large problems it may be the only feasiblemethod from the point of view of storage requirements.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

On the Application of a Generalization of Toeplitz Matrices to the Numerical Solution of Integral Equations with Weakly Singular Convolution Kernels

This paper discusses the numerical solution by product integrationof weakly singular Fredholm integral equations of the secondkind with symmetric difference kernels. The product integrationmethod uses a piecewise polynomial, in general, at most, continuousat the knots. A main result of the paper is to show that, owingto the difference kernel, a highly patterned linear system ofequations arises if the knots are equally spaced. Specificallythe order-N coefficient matrix is block-Toeplitz or a generalization,and centrosymmetric. An algorithm to solve this linear systemin O(N²) operations is presented. Unfortunately the asymptotic rate of convergence of the productintegration solution is limited if a uniform mesh is used. Amethod to improve the rate of convergence, while retaining apatterned coefficient matrix, is described. This method involvessubtraction of the singularities in the solution induced bythe weakly singular kernel. A higher rate of convergence forthe modified product integration method is shown. These resultsare illustrated by application to an integral equation arisingin outdoor sound propagation.     

Costing minimal-repair replacement policies over finite time horizons

An integral-equation technique is used to evaluate the expectedcost of maintaining a system functioning over the period (O,t] using two minimal-repair replacement policies. These costfunctions provide appropriate criteria to determine T^*, theoptimal scheduled replacement period over this finite time horizon.For both policies, it is shown that significant cost savingscan be achieved by using the T^* values predicted by the newmodels with a finite time horizon rather than those obtainedfrom the established asymptotic formulations. An adaptive finiteminimal-repair replacement policy is also formulated using dynamicprogramming, and the expected cost of this policy is shown tobe only slightly less than that of the best stationary policy.     

On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization

^** Email: bause{at}am.uni-erlangen.de Due to the increasing use of higher-order methods in computationalfluid dynamics, the question of optimal approximability of theNavier–Stokes equations under realistic assumptions onthe data has become important. It is well known that the regularitycustomarily hypothesized in the error analysis for parabolicproblems cannot be assumed for the Navier–Stokes equations,as it depends on non-local compatibility conditions for thedata at time t = 0, which cannot be verified in practice. Takinginto account this loss of regularity at t = 0, improved convergenceof the order (min{h^(5/2)–,h³/t^(1/4)+}), for any >0, is shown locally in time for the spatial discretization ofthe velocity field by (non-)conforming finite elements of third-orderapproximability properties. The error estimate itself is provedby energy methods, but it is based on sharp a priori estimatesfor the Navier–Stokes solution in fractional-order spacesthat are derived by semigroup methods and complex interpolationtheory and reflect the optimal regularity of the solution ast 0.     

Initial-layer theory and model equations of Volterra type

^** Email: amb16{at}nyu.edu It is demonstrated here that there exist initial layers to singularlyperturbed Volterra equations whose thicknesses are not of theorder of magnitude of O(), 0. It is also shown that the initial-layertheory is extremely useful because it allows one to constructthe approximate solution to an equation, which is almost identicalto the exact solution.     

Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model

In this paper, a semidiscrete finite element Galerkin methodfor the equations of motion arising in the 2D Oldroyd modelof viscoelastic fluids with zero forcing function is analysed.Some new a priori bounds for the exact solutions are derivedunder realistically assumed conditions on the data. Moreover,the long-time behaviour of the solution is established. By introducinga Stokes–Volterra projection, optimal error bounds forthe velocity in the L(L²) as well as in the L(H¹)-norms andfor the pressure in the L(L²)-norm are derived which are validuniformly in time t > 0.     

A Fast Direct Method for the Least Squares Solution of Slightly Overdetermined Sets of Linear Equations

Noble (1969) has described a method for the solution of N+Mlinear equations in N unknowns, which is based on an initialpartitioning of the matrix A, and which requires only the solutionof square sets of equations. He assumed rank (A) = N. We describehere an efficient implementation of Noble's method, and showthat it generalizes in a simple way to cover also rank deficientproblems. In the common case that the equation is only slightlyoverdetermined (M << N) the resulting algorithm is muchfaster than the standard methods based on M.G.S. or Householderreduction of A, or on the normal equations, and has a very similaroperation count to the algorithm of Cline (1973). Slightly overdetermined systems arise from Galerkin methodsfor non-Hermitian partial differential equations. In these systems,rank (A) = N and advantage can be taken of the structure ofthe matrix A to yield a least squares solution in (N²) operations.     

Global blow-up of separable solutions of the vorticity equation

In this paper we construct solutions to the equation on a finite interval in y which blow-up globallyin finite time. This equation arises in a number of physicalsituations and can be derived from the vorticity equation bylooking for stagnation-point type separable solutions for thetwo-dimensional streamfunction of the form xu(y, t). In theparticular application which has prompted the investigationreported in this paper, (*) is solved subject to boundary conditionsinvolving ²u/y². For this type of boundary condition the phenomenonof blow-up was first observed numerically by solving the initial-boundary-valueproblem for (*). These computations reveal that, depending onthe parameter combinations chosen, the solution to the initial-valueproblem may either blow-up globally in finite time or approacha steady state as t . Using the computations as a guide weconstruct the analytic behaviour of the solution close to theblow-up time using the methods of formal asymptotics.     

A Fast Galerkin Algorithm for Singular Integral Equations

A recent paper (Delves, 1977) described a variant of the Galerkinmethod for linear Fredholm integral equations of the secondkind with smooth kernels, for which the total solution timeusing N expansion functions is (N² ln N) compared with the standardGalerkin count of (N³). We describe here a modification of thismethod which retains this operations count and which is applicableto weakly singular Fredholm equations of the form where K₀(x, y) is a smooth kernel and Q contains a known singularity.Particular cases treated in detail include Fredholm equationswith Green's function kernels, or with kernels having logarithmicsingularities; and linear Volterra equations with either regularkernels or of Abel type. The case when g(x) and/or f(x) containsa known singularity is also treated. The method described yieldsboth a priori and a posteriori error estimates which are cheapto compute; for smooth kernels (Q = 1) it yields a modifiedform of the algorithm described in Delves (1977) with the advantagethat the iterative scheme required to solve the equations in(N²) operations is rather simpler than that given there.