Compact High-Order Finite Differences for Interface Problems in One Dimension

This paper is concerned with the construction and analysis ofcompact finite difference approximations to the model linearsource problem –(pu')' + qu = f where the functions p,q, and f can have jump discontinuities at a finite number ofpoints. Explicit formulae that give O(h²) O(h³) and O(h⁴) accuracyare derived, and a procedure for computing three-point schemesof any prescribed order of accuracy is presented. A rigoroustruncation and discretization error analysis is offered. Numericalresults are also given.     

Increased Accuracy Cubic Spline Solutions to Two-Point Boundary Value Problems

The relation between finite difference approximation and cubicspline solutions of a two-point boundary value problem for thedifferential equation y' +f(x)y^'+g(x)y = r(x) has been consideredin a previous paper. The present paper extends the analysisto the integral equation formulation of the problem. It is shownthat an improvement in accuracy (local truncation error O(h⁶)rather than O(h⁴)) now results from a cubic spline approximationand that for the particular case f(x) 0 the resulting recurrencerelations have a form and accuracy similar to the well-knownNumerov formula. For this case also a formula with local truncationerror O(h⁸) is derived.     

Notes on the Numerical Solution of the Biharmonic Equation

It is shown that the solution of the differenced form of theDirichlet problem for the biharmonic equation by iteration isachieved in O(h^–1)log h² steps of iteration where h isa step-size parameter from the mesh.     

On the Sloan iteration applied to integral equations of the first kind

When the piecewise constant collocation method is used to solvean integral equation of the first kind with logarithmic kernel,the convergence rate is O(h) in the L₂ norm. In this note weshow that O(h³) or O(h⁵) convergence in any Sobolev norm (andthus, for example, in L) may be obtained by a simple cheap postprocessingof the original collocation solution. The construction of thepostprocessor is based on writing the first kind equation asa second kind equation, and applying the Sloan iteration tothe latter equation. The theoretical convergence rates are verifiedin a numerical example.     

The Fast Numerical Solution of Very Large Elliptic Difference Schemes

Let L_kv_k = g_k be a system of difference equations discretizingan elliptic boundary value problem. Assume the system to be"very large", that means that the number of unknowns exceedsthe capacity of storage. We present a method for solving theproblem with much less storage requirement. For two-dimensionalproblems the size of the needed storage decreases from O(h^–2)to (or even O(h^–5/4)). The computational work increasesonly by a factor about six. The technique can be generalizedto nonlinear problems. The algorithm is also useful for computerswith a small number of parallel processors.     

Error Analysis of the Enthalpy Method for the Stefan Problem

In this paper an error bound is derived for a practical piecewiselinear finite-element approximation of an enthalpy formulationof the multidimensional Stefan problem with an implicit timediscretization. It is shown that if the time step t is O(h),then the error in the temperature measured in the L² norm isO(h).     

A New Fourth-order Finite-difference Method for Computing Eigenvalues of Fourth-order Two-point Boundary-value Problems

We present a new finite-difference method for computing eigenvaluesof two-point boundary-value problems involving a fourth-orderdifferential equation. Our finite-difference method leads toa generalized seven-diagonal symmetric-matrix eigenvalue problemand provides O(h⁴)-convergent approximations for the eigenvalues.     

Optimal Recovery in the Finite-Element Method, Part 2: Defect Correction for Ordinary Differential Equations

This paper continues the study of optimal recovery formulaefor piecewise linear approximations to an unknown function u(x):here recovery is with respect to the mixed norm which occurs naturally when u arises from a Sturm-Liouvilleproblem. It is shown that local recovery is possible only whenp = O(qh²) Otherwise, global recovery is required and this isachieved by a defect-correction approach, closely related tothe difference-correction techniques of Fox used with finite-differencemethods. Results on the ensuing improvement of accuracy aregiven which show that typically O(h⁴) can be achieved with onesimple iteration.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.     

End Conditions for Interpolatory Quintic Splines

Present address: Department of Mathematics, University of Tabriz, Tabriz, Iran. Accurate end conditions are derived for quintic spline interpolationat equally spaced knots. These conditions are in terms of availablefunction values at the knots and lead to O(h⁶) convergence uniformlyon the interval of interpolation.     

Implicit finite difference schemes for a linear model of well-reservoir coupling

Standard reservoir models usually consider wells as Dirac measuresover an interval length. Moreover, the well-reservoir couplingis taken into account under quite simplified assumptions. Mostrecently, however, attention has been drawn to the fact thatin some situations, such as those related to non-vertical wells,these simplifications do not allow us to model some relevantmechanisms of the coupled flow. Therefore, more complex alternativemodels have been proposed recently in the oil reservoir simulationliterature. A linearized version for the well-reservoir couplingcan be written, in an appropriate functional setting, in theform U'(t) + AU(t) = F(t). In this work we discuss implicitin time discretizations of this equation, of the form { Uⁿ⁺¹_h - Uⁿ_h÷+A_hUⁿ⁺¹_h = Fⁿ⁼¹_h, U⁰_h=U_0.h We propose two different approximations, corresponding to first-and second-order spatial truncation errors, and we establishthe convergence of both approximations.     

On the Solution of Some Elliptic Difference Equations

Iterative methods for the solution of some nonlinear ellipticdifference systems, approximating the first boundary value problemare considered. If h > 0 is the network step in the spaceof variables x = (x₁, x₂,..., x_p) and 2m is the order of theoriginal boundary value problem, then the iterative methodsproposed give solution of accuracy with the expenditure ofO(|In | h^–(p+m–)) and O(|In | |In h| h^–p)arithmetic operations in the case of a general region and arectangular parallelepiped respectively. In the case p = 2 theestimate O(|In | h^{–[2+ (m/2)]}) is obtained if the regionis made up of rectangles with sides parallel to the co-ordinateaxes.     

Convergence of a fully discrete approximation for advected mean curvature flows

Let (t) be a closed curve in R² which propagates in its normaldirection n with velocity V = --q.n-g, where is the mean curvatureof (t) and g and q are given represent, respectively, a forcingterm and a vector field. In this paper we prove that such flowscan be approximated by numerical solutions of advection Allen-Cahnequations. It is shown that the zero level set of the fullydiscrete solution using explicit time stepping converges evenpast singularities to the true interface provided that no fatteningoccurs and , h² O(⁴), where h and denote the mesh size andthe time step. For smooth flows an optimal O(²)-rate of convergenceis derived provided , h² O(⁵). The analysis is based on constructingfully discrete barriers via an explicit parabolic projectionand Lipschitz dependence of the viscosity solutions with respectto perturbations of data.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

On the Effect of Boundary Conditions in Collocation by Polynomial Splines for the Solution of Boundary Value Problems in Ordinary Differential Equations

Consider the numerical solution of a boundary-value problemfor a differential equation of order m using collocation ofa polynomial spline of degree n m on a uniform mesh of sizeh. We describe several collocation schemes which differ onlyin the boundary collocation conditions and which include a "natural"spline collocation scheme. Taking account of derived asymptoticerror bounds most of which are, roughly speaking, of O(hn^12m+1), we discuss the computational effectiveness of the variousschemes.     

Total Flux Estimates for a Finite-Element Approximation of Elliptic Equations

An elliptic boundary-value problem on a domain with prescribedDirichlet data on _I is approximated using a finite-elementspace of approximation power h^K in the L² norm. It is shownthat the total flux across _I can be approximated with an errorof O(h^K) when is a curved domain in Rⁿ (n = 2 or 3) and isoparametricelements are used. When is a polyhedron, an O(h^2K–2)approximation is given. We use these results to study the finite-elementapproximation of elliptic equations when the prescribed boundarydata on _I is the total flux. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton, Sussex BN1 9QH.     

Orbits in the Maximal Ideal Space of H{infty}

Let H be the Banach algebra of bounded analytic functions inthe open unit disc D. We can define the rotation in the maximalideal space M(H). For a point x in M(H)\D, an orbit O(x) isnot closed in M(H). It is proved that there exists a point xin M(H) such that x is not contained in the Shilov boundaryX and cl O(x), the closure of O(x), contains X, and there existsa point y in M(H)\(D X) such that cl O(y) X. The rotationpresents many problems concerning H. The purpose of this paperis to discuss these problems.     

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.     

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.