Showing papers on "Numerical analysis published in 1987"

Moduli of smoothness

Abstract: The book introduces a new way of measuring smoothness. The need for this new concept arises from the failure of the classical moduli of smoothness to solve some basic problems, such as characterizing the behaviour of best polynomial approximation in Lp -1,1 . The new modulus, which has a simple form, can also be described as a Peetre K functional between an Lp space and a weighted Sobolev space. Connections between interpolation of spaces and approximation theory are utilized in applying the modulus of smoothness. The applications include best (weighted) polynomial approximation on a finite interval, characterization of the rate of approximation given by classical operator processes such as Bernstein, Kantorovich, Szasz-Mirakjan, and Post-Widder operators, Freud-type weighted polynomial approximation on infinite intervals with exponentially decreasing weights and polynomial approximation in several variables. Special emphasis is placed on the computability aspect of the moduli. The results are new, and complete proofs are given. It is hoped that the book will be of interest and useful for mathematicians working in approximation theory, interpolation of spaces, numerical analysis and real analysis.

An efficient newton‐like method for molecular mechanics energy minimization of large molecules

Abstract: Techniques from numerical analysis and crystallographic refinement have been combined to produce a variant of the Truncated Newton nonlinear optimization procedure. The new algorithm shows particular promise for potential energy minimization of large molecular systems. Usual implementations of Newton's method require storage space proportional to the number of atoms squared (i.e., O(N2)) and computer time of O(N3). Our suggested implementation of the Truncated Newton technique requires storage of less than O(N1.5) and CPU time of less than O(N2) for structures containing several hundred to a few thousand atoms. The algorithm exhibits quadratic convergence near the minimum and is also very tolerant of poor initial structures. A comparison with existing optimization procedures is detailed for cyclohexane, arachidonic acid, and the small protein crambin. In particular, a structure for crambin (662 atoms) has been refined to an RMS gradient of 3.6 × 10−6 kcal/mol/A per atom on the MM2 potential energy surface. Several suggestions are made which may lead to further improvement of the new method.

The numerical analysis of ordinary differential equations

Abstract: A temperature control system for individual rooms within a home is disclosed having a pair of thermostatically controlled switches settable to "occupied" and "unoccupied" temperatures respectively with a clock controlled switching arrangement for determining which thermostatically controlled switch is effective during predetermined hours of the day. The clock selected thermostatically controlled switch is coupled by way of a relay to control a forced air duct opening and closing louver arrangement so that the duct may be opened for either heating or cooling during unoccupied periods of time a lesser percentage than it is opened during periods of the day when the room is typically occupied.

A high-order spectral method for the study of nonlinear gravity waves

Abstract: We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number ( N = O (1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M , and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness ( ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

One-step and extrapolation methods for differential- algebraic systems

Abstract: The paper analyzes one-step methods for differential-algebraic equations (DAE) in terms of convergence order. In view of extrapolation methods, certain perturbed asymptotic expansions are shown to hold. For the special DAE extrapolation solver based on the semi-implicit Euler discretization, the perturbed order pattern of the extrapolation tableau is derived in detail. The theoretical results lead to modifications of the known code. The efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion.

A new numerical method for surface hydrodynamics

Abstract: We present a new numerical method for studying the evolution of free and bound waves on the nonlinear ocean surface. The technique, based on a representation due to Watson and West (1975), uses a slope expansion of the velocity potential at the free surface and not an expansion about a reference surface. The numerical scheme is applied to a number of wave and wave train configurations including longwave-shortwave interactions and the three-dimensional instability of waves with finite slope. The results are consistent with those obtained in other studies. One strength of the technique is that it can be applied to a variety of wave train and spectral configurations without modifying the code.

Sinc Methods for Quadrature and Differential Equations

Abstract: 1. Preliminary material 2. Numerical methods on the real line 3. Numerical methods on an arc 'Gamma' 4. The Sinc-Galerkin method 5. Steady problems 6. Time-dependent problems Appendix A. Linear algebra References.

On the solution of integral equations with strongly singular kernels

Abstract: Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

Explicit Runge-Kutta (-Nystro¨m) methods with reduced phase errors for computing oscillating solutions

Abstract: We construct explicit Runge–Kutta (–Nystrom) methods for the integration of first (and second) order differential equations having an oscillatory solution. Special attention is paid to the phase errors (or dispersion) of the dominant components in the numerical oscillations when these methods are applied to a linear, homogeneous test model. RK(N) methods are constructed which are dispersive of orders up to 10, whereas the (algebraic) order of accuracy is only 2 or 3. Application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced.

Solution of Partial Differential Equations on Vector and Parallel Computers

Abstract: In this work we review the present status of numerical methods for partial differential equations on vector and parallel computers. A discussion of the relevant aspects of these computers and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial-boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. A brief discussion of application areas utilizing these computers is included.

Numerical analysis of planar optical waveguides using matrix approach

Abstract: We present here a simple matrix method for obtaining propagation characteristics, including losses for various modes of an arbitrarily graded planar waveguide structure which may have media of complex refractive indices. We show the applicability of the method for obtaining leakage losses and absorption losses, as well as for calculating beat length in directional couplers. The method involves straightforward 2 × 2 matrix multiplications, and does not require the solutions of any transcendental or differential equations.

Numerical investigation of the three-dimensional development in boundary-layer transition

Abstract: A numerical method for solving the complete Navier-Stokes equations for incompressible flows is introduced that is applicable for investigating three-dimensional transition phenomena in a spatially growing boundary layer. Results are discussed for a test case with small three-dimensional disturbances for which detailed comparison to linear stability theory is possible. The validity of our numerical model for investigating nonlinear transition phenomena is demonstrated by realistic spatial simulations of the experiments by Kachanov and Levchenko1 for a subharmonic resonance breakdown and of the experiments of Klebanoff et al.2 for a fundamental resonance breakdown.

Large scale scientific computing

Abstract: I: Initial Value Problems for Ode's and Parabolic Pde's.- 1. Algorithms for Semiconductor Device Simulation.- 2. Hierarchical Bases in the Numerical Solution of Parabolic Problems.- 3. Extrapolation Integrators for Quasilinear Implicit ODE's.- 4. Numerical Problems Arising from the Simulation of Combustion Phenomena.- 5. Numerical Computation of Stiff Systems for Nonequilibrium.- 6. Finite Element Simulation of Saturated-Unsaturated Flow Through Porous Media.- II: Boundary Value Problems for ODE's and Elliptic PDE's.- 7. Numerical Pathfollowing Beyond Critical Points in ODE Models.- 8. Computing Bifurcation Diagrams for Large Nonlinear Variational Problems.- 9. Extinction Limits for Premixed Laminar Flames in a Stagnation Point Flow.- 10. A Numerical Method for Calculating Complete Theoretical Seismograms in Vertically Varying Media.- 11. On a New Boundary Element Spectral Method.- III: Hyperbolic PDE's.- 12. A High Order Non-Oscillatory Shock Capturing Method.- 13. Vortex Dynamics Studied by Large-Scale Solutions to the Euler Equations.- IV: Inverse Problems.- 14. Numerical Backprojection in the Inverse 3D Radon Transform.- 15. A Direct Algebraic Algorithm in Computerized Tomography.- 16. A Two-Grid Approach to Identification and Control Problems for Partial Differential Equations.- V: Optimization and Optimal Control Problems.- 17. Solving Large-Scale Integer Optimization Problems.- 18. Numerical Treatment of State & Control Constraints in the Computation of Feedback Laws for Nonlinear Control Problems.- 19. Optimal Production Scheme for the Gosau Hydro Power Plant System.- VI: Algorithm Adaptation on Supercomputers.- 20. The Use of Vector and Parallel Computers in the Solution of Large Sparse Linear Equations.- 21. Local Uniform Mesh Refinement on Vector and Parallel Processors.- 22. Using Supercomputer to Model Heat Transfer in Biomedical Applications.- Speakers.

Numerical simulations of nonlinear axisymmetric flows with a free surface

Abstract: A numerical method is developed for nonlinear three-dimensional but axisymmetric free-surface problems using a mixed Eulerian-Lagrangian scheme under the assumption of potential flow. Taking advantage of axisymmetry, Rankine ring sources are used in a Green's theorem boundary-integral formulation to solve the field equation; and the free surface is then updated in time following Lagrangian points. A special treatment of the free surface and body intersection points is generalized to this case which avoids the difficulties associated with the singularity there. To allow for long-time simulations, the nonlinear computational domain is matched to a transient linear wavefield outside. When the matching boundary is placed at a suitable distance (depending on wave amplitude), numerical simulations can, in principle, be continued indefinitely in time. Based on a simple stability argument, a regriding algorithm similar to that of Fink & Soh (1974) for vortex sheets is generalized to free-surface flows, which removes the instabilities experienced by earlier investigators and eliminates the need for artificial smoothing. The resulting scheme is very robust and stable.For illustration, three computational examples are presented: (i) the growth and collapse of a vapour cavity near the free surface; (ii) the heaving of a floating vertical cylinder starting from rest; and (iii) the heaving of an inverted vertical cone. For the cavity problem, there is excellent agreement with available experiments. For the wave-body interaction calculations, we are able to obtain and analyse steady-state (limit-cycle) results for the force and flow field in the vicinity of the body.

Adaptive grid generation

Abstract: The fundamental principles of adaptive grid generation for the numerical analysis of physical phenomena described by systems of partial differential equations are examined in an analytical review. Topics addressed include weight functions, equidistribution in one dimension, the specification of coefficients in the linear weight, the attraction to a given grid on a curve, evolutionary forces, and metric notation. Consideration is given to curve-by-curve methods, finite-volume methods, variational methods, and temporal aspects.

Nonlinear Methods in Numerical Analysis

Abstract: I. Continued Fractions. Since these play an important role, the first chapter introduces their basic properties, evaluation algorithms and convergence theorems. From the section dealing with convergence it can be seen that in certain situations nonlinear approximations are more powerful than linear approximations. The recent notion of branched continued fraction is introduced in the multivariate section and is later used for the construction of multivariate rational interpolants. II. Pade Approximants. A survey of the theory of these local rational approximants for a given function is presented, including the problems of existence, unicity and computation. Also considered are the convergence of sequences of Pade approximants and the continuity of the Pade operator which associates with a function its Pade approximant of a certain order. A special section is devoted to the multivariate case. III. Rational Interpolants. These rational functions fit a given function at some given points. Many results of the previous chapter remain valid for this more general case where the interpolation conditions are spread over several points. In between the rational interpolation case and Pade approximation case lies the theory of rational Hermite interpolation where each interpolation point can be assigned more than one interpolation condition. Some results on the convergence of sequences of rational Hermite interpolants are mentioned and multivariate rational interpolants are introduced in two different ways. IV. Applications. The previous types of rational approximants are used here to develop several numerical methods for the solution of classical problems such as convergence acceleration, nonlinear equations, ordinary differential equations, numerical quadrature, partial differential equations and integral equations. Many numerical examples illustrate the different techniques, and it is seen that nonlinear methods are very useful in situations involving singularities. Subject Index.

Stability analysis of slopes with general nonlinear failure criterion

Abstract: The upper bound method of limit analysis of perfect plasticity is applied to stability problems of slopes with a general nonlinear failure criterion. Based on the upper bound method, a numerical procedure is suggested, which converts the complex system of differential equations to an initial value problem. Using this numerical procedure, an effective numerical method, called the inverse method, suitable for the solution of slope stability problems in soil mechanics with a general nonlinear failure criterion, is presented. A general nonlinear failure criterion for soils is also suggested, from which the effects of nonlinear failure parameters on the stability of slopes are discussed.

Constrained Differential Optimization

Abstract: Many optimization models of neural networks need constraints to restrict the space of outputs to a subspace which satisfies external criteria Optimizations using energy methods yield "forces" which act upon the state of the neural network The penalty method, in which quadratic energy constraints are added to an existing optimization energy, has become popular recently, but is not guaranteed to satisfy the constraint conditions when there are other forces on the neural model or when there are multiple constraints In this paper, we present the basic differential multiplier method (BDMM), which satisfies constraints exactly; we create forces which gradually apply the constraints over time, using "neurons" that estimate Lagrange multipliers The basic differential multiplier method is a differential version of the method of multipliers from Numerical Analysis We prove that the differential equations locally converge to a constrained minimum Examples of applications of the differential method of multipliers include enforcing permutation codewords in the analog decoding problem and enforcing valid tours in the traveling salesman problem

The Wrinkling of Thin Membranes: Part II—Numerical Analysis

Abstract: Using a weighted residual method, a geometrically and physically nonlinear membrane element is derived, which can be used in the analysis of anisotropic membranes. What is special about the formulation is that the wrinkling behavior of the element is incorporated. If wrinkling occurs the stress situation in the element is determined by making use of a modified deformation tensor. A structure may have completely slack regions, leading to a singular stiffness matrix. Because of this we have chosen to use a restricted step method for the iterative solution procedure. A simple shear test is used to compare numerical and analytical results which show good agreement.

On the Numerical Approximation of Phase Portraits Near Stationary Points

Abstract: We show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one-step methods or multi-step methods satisfying a strong root condition. This means that any continuous trajectory can be approximated by an appropriate discrete trajectory, and vice versa, to the correct order of convergence and uniformly on arbitrarily large time intervals. In particular, the stable and unstable manifolds of the discretization converge to their continuous counterparts.

Finite element solution of nonlinear elliptic problems

Abstract: The study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed Dirichlet-Neumann boundary conditions is presented. In the discretization variational crimes are commited (approximation of the given domain by a polygonal one, numerical integration). With the assumption that the corresponding operator is strongly monotone and Lipschitz-continuous and that the exact solutionu∈H1(Ω), the convergence of the method is proved; under the additional assumptionu∈H2(Ω), the rate of convergenceO(h) is derived without the use of Green's theorem.

Comparison of the FFT Conjugate Gradient Method and the Finite-Difference Time-Domain Method for the 2-D Absorption Problem

Abstract: The need for high-resolution distributive dosimetry demands a numerical method capable of handling finely discretized, arbtrarily inhomogeneous models of biological bodies At present, two of the most promising methods in terms of numerical efficiency are the fast-Fourier-transform conjugate gradient method (FFT-CGM) and the finite-difference time-domain (FD-TD) method In this paper, these two methods are compared with respect to their ability to solve the 2-D Iossy dielectric cylinder problem for both the TM and TE incident polarizations Substantial errors are found in the FFT-CGM solutions for the TE case The source of these errors is explained and a modified method is developed which, although inefficient, alleviates the problem and illuminates the difficulties encountered in applying the pulse-basis method of moments to biological problems In contrast, the FD-TD method is found to yield excellent solutions for both polarizations This, coupled with the numerical efficiency of the FD-TD method, suggests that it is superior to the FFT-CGM for biological problems

Error analysis of the bjo¨rck-pereyra algorithms for solving vandermonde systems

Abstract: A forward error analysis is presented for the Bjorck-Pereyra algorithms used for solving Vandermonde systems of equations. This analysis applies to the case where the points defining the Vandermonde matrix are nonnegative and are arranged in increasing order. It is shown that for a particular class of Vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only, being independent of the condition number of the coefficient matrix. By comparing appropriate condition numbers for the Vandermonde problem with the forward error bounds it is shown that the Bjorck-Pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right-hand side vector on the computer. A technique for computing “running” a posteriori error bounds is derived. Several numerical experiments are presented, and it is observed that the ordering of the points can greatly affect the solution accuracy.

Some numerical methods for the computation of capillary free boundaries governed by the Navier-Stokes equations

Abstract: In §, the trial method, Newton’s method, and the total linearization method are applied to a simple one-dimensional free boundary (FB) problem for which the solution is known. Attention will be focused on convergence speed and computer implementation facilities of these methods. Section 2, discusses static FB problems in two- and three-dimensional containers. These FBs are governed by the Laplace–Young equation. In §3, two stationary FB problems governed by the Navier–Stokes equations are formulated and these problems are solved using the three methods introduced in §1. A thermocapillary FB problem governed by the Navier–Stokes equations coupled with the heat equation is studied in §4. In §5, a moving FB problem governed by the Navier–Stokes equations will be considered within the context of perturbation theory. Finally, in §6, some remarks are made on present and future research in the field of capillary FB problems governed by the Navier–Stokes equations.