Newton Iteration Process Research Articles

A method for constructing generalized Runge-Kutta methods

In the implementation of an implicit Runge-Kutta formula, we need to solve systems of nonlinear equations. In this paper, we analyze the Newton iteration process and a modified Newton iteration process for solving these equations. Then we propose the methods in which we take only a fixed finite number of iterations and adopt the last iterate as the approximate solution for the nonlinear equations. Our methods are a kind of generalized Runge-Kutta methods with the same order as the original Runge-Kutta formula and inherit its linear stability properties of the original implicit Runge-Kutta formula, for example, AN-stability, L-stability and S-stability. Based on this fact, we construct three methods of order five imbedding a fourth-order formula for error estimation. Finally, test results for 25 stiff problems are discussed.

Dynamical-system formulation of the eigenvalue moment method

The eigenvalue moment method (EMM), a linear-programming (LP) based technique for generating converging bounds to quantum eigenenergies, is reformulated as an iterative dynamical system (DS). Important convexity properties are uncovered significantly impacting the theoretical and computational implementation of the EMM program. In particular, whereas the LP-based EMM formulation (LP-EMM) can require the generation and storage of many inequalities [up to several thousand for a 10-missing moment problem (${\mathit{m}}_{\mathit{s}}$=10)], the dynamical-system formulation (DS-EMM) generates a reduced set of inequalities [of order O(${\mathit{m}}_{\mathit{s}}$+1)]. This is made possible by replacing the LP generation of deep interior points (DIP's) by a Newton iteration process. The latter generates an optimal set of DIP's sufficient to determine the existence or nonexistence of the relevant missing moment polytopes. The general DS-EMM theory is presented together with numerical examples.

A Newton iteration process for inverse eigenvalue problems

We suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases. For the numerical solution of this problem we propose a Newton iteration process and compare it with a known method. Finally we apply it to a numerical example.

Structure and stress in the gas–liquid–solid contact region

The physics of the region where a fluid–fluid interface meets a solid, a region where wetting phenomena are governed, is studied with the gradient theory of an inhomogeneous equilibrium fluid. The results illuminate the molecular origins of contact angle and relate this quantity to local fluid structure and stress.Liquid and vapour in equilibrium between two flat, parallel walls is the system examined. The distance between the walls is varied to study the effect of confinement on the contact region. The fluid is a one-component, Peng–Robinson fluid. The solid–fluid potential model is based on the Lennard-Jones 6–12 molecular pair potential.The contact region is divided into subdomains and the density field is expanded in finite-element basis functions. On each subdomain the Galerkin weighted residual of the gradient equation is put equal to zero and the resulting set of non-linear algebraic equations is solved by Newton's iteration process. The converged Jacobian from Newton's process provides information about parametric sensitivity and thermodynamic stability. Asymptotic approximations are used wherever possible to enhance the power of the basis set.The primary results are density and stress profiles. Young's equation of capillarity is found to be valid when properly interpreted. There is, however, a contact region near the solid where Young's equation is not valid and where the fluid–fluid meniscus loses its meaning. Moreover, in the small systems for which theoretical or molecular dynamics calculations are presently affordable the angle of inclination of the meniscus is a function of distance from the wall and therefore cannot be identified with Young's contact angle.Besides their value to surface and colloid science the results testify to the power of modern computer-aided analysis by finite-element subdomain methods.

The Initial-Value Adjusting Method for Problems of the Least Squares Type of Ordinary Differential Equations

^ j=i where tj are given points on [a, fo], a = tl<t2<--<tN = b, and Lj and dj are given matrices and vectors, respectively. Some authors have discussed these types of problems and proposed several numerical methods to solve them. Banks and Groome [I] have discussed the quasilinearization algorithm and established the quadratic convergence of it. Urabe [8] has shown that the problem can be reduced to the multipoint boundary value problem of nonlinear boundary condition, and proposed the application of the Newton iterative process. After Urabe's works, Fujii [2] has shown another reduction of the problem to a boundary value problem and discussed the Chebyshev-series-approximation with a-posteriori error bound. In the present paper, we first propose a new method which is an applied version of the initial-value adjusting method given by Ojika and Kasue [6] for

The transition problem in pumped aquifers

Confined aquifers may develop water table conditions near the well bore and under prolonged pumping stresses this condition may spread to the recharge area. This situation is defined as the transition problem. The physics of the transition problem is examined by modeling well flow with a finite difference computer simulator. The computer simulator is based on two‐dimensional, two‐phase flow and takes into consideration formation compressibility, fluid (air and water) compressibility, solubility of air in water and a seepage face boundary condition at the well bore. A fully implicit finite difference scheme is used to solve the nonlinear flow equations by employing a modified Newton iteration process. The results obtained in this study agree with field data. Current analytical approaches seem to overestimate drawdown levels in time. The analysis shows that the transition problem is a unique combination of confined and unconfined flow. Gravity flow near the well bore appears to dominate the total flow pattern and affects drawdown at farther distances from the well bore where flow is radial. It was also shown that partially penetrating wells behave much like fully penetrating wells at certain radial distances from the well bore.

Nonequilibrium hypersonic flow of carbon dioxide gas over blunt bodies

The Navier-Stokes equations are used to investigate hypersonic flow of carbon dioxide gas over blunt bodies with allowance for nonequilibrium development of chemical reactions and vibrational relaxation of the CO2 molecules. The problem is solved by the method of stabilization by means of an implicit difference scheme that includes the use of Newton's iterative process. The results are given of calculations of the flow field, the convective heat flux, and the frictional stresses on the surface of blunt cones with spherical noses. The influence of admixtures on the flow field and the heat fluxes is investigated. The results of the calculations are compared with the locally self-similar solution for the neighborhood of the front stagnation point.

Effective Price Mechanisms

It is known that the price mechanism whereby the rate of change of a price is proportional to the excess demand of the corresponding commodity need not converge to a competitive equilibrium for a pure exchange economy with more than two commodities. On the other hand, there exist convergent price mechanisms, similar to the Newton iterative process, where the rate of change of the prices is determined by the excess demand and the marginal excess demands of all the commodities. This is a considerable informational requirement. It is shown that this requirement cannot be substantially reduced for any convergent price mechanisms, that is for price mechanisms expressed in terms of a difference or differential equation where the solutions converge to a competitive equilibrium.

Mechanism of cyclopropane-propene isomerization

A SINDO method was used to investigate the sequence of rearrangements in the cyclopropane-propene isomerization. Geometries of reactant and product were optimized for a complete set of 21 independent internal coordinates. Then the bond angle of the migrating H atom was chosen as reaction coordinate and all other geometrical parameters were energy minimized along this reaction coordinate. In the neighborhood of the transition state a Newton type iteration process is initiated to locate the transition state. The latter is characterized by a CCC bond angle close to 90 ° and an almost finished migration of the H atom. The activation energy was found about 25% lower than the experimental value of 65 kcal/mole. A sequence of rearrangements was established, suggesting the initiation of the reaction by a combination of two normal modes: a torsional vibration of the methylene groups together with an asymmetric stretch of the carbon framework. Our calculations classify the reaction as concerted.

Optimum Simulation Of B-H Curves In FEMAnalysis Of Magnetic Fields

The paper introduces a new algorithm for the numerical simulation of magnetisation curves, suitable for Newton iterative process in FEM analysis of non-linear magnetic fields, both for dc magnetisation curves and for hysteresis loops. The algorithm, from the complete data sheet of the magnetic material, determines the number and the location of the points which set is minimum and ensures a global error lower that a control quantity.

Non‐linear Transformations of Divergent and Slowly Convergent Sequences

This paper discusses a family of non‐linear sequence‐to‐sequence transformations designated as ek, ekm, ẽk, and ed. A brief history of the transforms is related and a simple motivation for the transforms is given. Examples are given of the application of these transformations to divergent and slowly convergent sequences. In particular the examples include numerical series, the power series of rational and meromorphic functions, and a wide variety of sequences drawn from continued fractions, integral equations, geometry, fluid mechanics, and number theory. Theorems are proven which show the effectiveness of the transformations both in accelerating the convergence of (some) slowly convergent sequences and in inducing convergence in (some) divergent sequences. The essential unity of these two motives is stressed. Theorems are proven which show that these transforms often duplicate the results of well‐known, but specialized techniques. These special algorithms include Newton's iterative process, Gauss's numerical integration, an identity of Euler, the Padé Table, and Thiele's reciprocal differences. Difficulties which sometimes arise in the use of these transforms such as irregularity, non‐uniform convergence to the wrong answer, and the ambiguity of multivalued functions are investigated. The concepts of antilimit and of the spectra of sequences are introduced and discussed. The contrast between discrete and continuous spectra and the consequent contrasting response of the corresponding sequences to the e1 transformation is indicated. The characteristic behaviour of a semiconvergent (asymptotic) sequence is elucidated by an analysis of its spectrum into convergent components of large amplitude and divergent components of small amplitude.