Quadratic Schemes Research Articles

A Spline Difference Scheme on a Piecewise Equidistant Grid

AbstractWe consider singularly perturbed boundary value problems of reaction‐diffusion type and their discretization via quadratic spline difference schemes on a piecewise equidistant mesh of the Shishkin type. On such a mesh we prove that a solution to the discretisation is almost second order accurate in the discrete maximum norm, uniformly in the perturbation parameter. Numerical results are presented, which verify this rate of convergence.

First principles determination of magnetocrystalline anisotropy for surfaces and interfaces using a torque method (abstract)

Perpendicular magnetic alignment is vital for high density magneto-optical recording materials. Despite the tremendous theoretical/computional advances made during the last few decades, the determination of magnetocrystalline anistropy (MCA) from first principles still remains a great challenge for complex systems. We will describe our recently proposed torque method for the first principles determination of MCA. In the usual first principles methods, one calculates the band energies associated with two magnetization directions and substracts one from the other. Within this approach, one has the difficulty of getting rid of the random fluctuations arising from the two different Fermi surfaces due to different magnetization directions. We show that to accurately determine the spin-orbit induced uniaxial ansisotropy energy for surfaces/interfaces, calculation of the torque at a specific angle is sufficient and one avoids the complexities associated with two Fermi surfaces by employing the Feynman-Hellman theorem. In the k-space integrations, we used both linear and quadratic interpolation schemes and convergence is assured when these two schemes agree to the accuracy desired. Examples, including Fe and Co multilayer systems, will be presented to demonstrate the efficiency and precision of this method. Detailed comparisons with previously proposed state-tracing method by Wang et al. are also made and discussed.

Impacts of model improvements on general circulation model sensitivity to sea‐surface temperature forcing

AbstractThe sensitivity of the 4° × 5° resolution general circulation model of the Goddard Institute for Space Studies to some basic changes in model formulation was studied by comparing parallel simulations forced by global sea‐surface temperature of June‐August 1987 and 1988. The new modelling schemes were substituted for the control model's parameterizations of moist convection, planetary boundary layer, ground hydrology, and cloud optical thickness in a series of sensitivity experiments. In addition, linear and quadratic upstream schemes for advecting tracers were tried in place of second‐order differencing. Elimination of the vertical mixing of horizontal momentum by moist convection was also tested. Impacts of the new modelling schemes on simulated circulation, temperature, and precipitation rates were inferred from pairs of simulations made by model versions that differed with respect to a single change. No discernible positive impacts were found for the new ground hydrology scheme or for changes in the determination of cloud optical thickness. Profiles of zonal wind speeds and lower tropospheric circulation patterns, both in the tropics, were more realistic when the vertical mixing of horizontal momentum was included. Major improvements in modelling the interannual variability of the planetary circulation, mid‐tropospheric temperature, and precipitation can be attributed to the salutory effects of the new moist convection, new planetary boundary layer, and the quadratic upstream scheme for advecting tracers.

Solutions of nonstandard nth order initial value problems

Differential equations of the form y ( n) = f ( t, y, y',…, y ( n) , where f is not necessarily linear in its arguments, represent certain physical phenomena (e.g., free oscillations of positively damped systems, nonlinear systems subject to harmonic excitations, motion of a particle on a rotating parabola) and have been known for quite some time. Earlier we established the existence of a unique solution of the nonstandard first order initial value problem y' = f ( t, y, y'), y( t 0) = y 0 under certain natural hypotheses on f and developed some linear and quadratic convergent numerical schemes for the construction of approximate solution of the above problem. In this paper we establish existence results and develop some linearly convergent numerical schemes for the construction of solution of nonstandard nth order initial value problems, and we solve two physical examples.

Numerical aspects of calculation of confined swirling flows with internal recirculation

AbstractPredictions were performed for two different confined swirling flows with internal recirculation zones. The convection terms in the elliptic governing equations were discretized using three different finite differencing schemes: hybrid, quadratic upwind interpolation and skew upwind differencing. For each flow case, calculations were carried out with these schemes and successively refined grids were employed. For the turbulent flow case the k‐ε turbulence model was used. The predicted cases were a laminar swirling flow investigated by Bornstein and Escudier, and a turbulent low‐swirl case studied by Roback and Johnson. In both cases an internal recirculation zone was present. The laminar case is well predicted when account is taken of the estimated radial velocity component at the chosen inlet plane. The quadratic upwind interpolation and skew upwind schemes predict the main features of the internal recirculation zone also with a coarse grid. The turbulent case is well predicted with the coarse as well as the finer grids, the skew upwind and quadratic upwind interpolation schemes yielding results very close to the measurements. It is concluded that the skew upwind scheme reaches grid independence slightly before the quadratic upwind scheme, both considerably earlier than the hybrid scheme.

Ionic Thermocurrent Measurements Following Different Schemes of Heating

Equations for the depolarization current recorded following linear, reciprocal, quadratic and square root schemes of heating in ionic thermocurrent measurements have been derived. Methods for the evaluation of the dielectric relaxation parameters following the proposed schemes of heating have also been developed.

Indirect adaptive techniques for fixed controller performance enhancement

Abstract This paper develops the adaptive disturbance estimate feedback schemes of a companion paper for enhancing the performance of controllers designed by off-line techniques. The developments are based on the parametrization theory for the class of all stabilizing controllers for a nominal plant, and the dual class of plants stabilized by a nominal controller. Such parametrization allows us conveniently to parametrize plant uncertainties for on-line identification and control purposes, minimizing the effects of unmodelled dynamics. Based on these parametrizations, along with prefiltering which minimizes the effect of unmodelled dynamics, standard adaptive stabilization, adaptive pole assignment, or adaptive linear quadratic schemes are shown to achieve controller enhancement. The idea is to exploit a priori information about a plant and design objectives in an off-line design, and yet exploit the power of adaptive techniques to learn and tune on-line. Attention is focused on techniques for fixed but u...

Quadratic form schemes and quaternionic schemes

Quadratic form schemes and quaternionic schemes

On the design of incentive schemes under moral hazard and adverse selection

This paper characterizes a class of optimal incentive schemes in a simple principal–agent model which allows for moral hazard and adverse selection. We show that incentive compatible allocation can always be (approximately) implemented through a menu of quadratic incentive schemes. It is also proved that the set of incentive compatible allocations is independent of the distribution of the additive uncertainty which affects the outcome. Informational requirements and economic interpretation of quadratic and linear schemes are discussed.

Image Segmentation by semantic method

The problem of region detection is addressed. Linear and quadratic approximation schemes are used to approximate the regions in an image. A set of attributes, which represent the properties of a region, are defined. A distance function, which has structural as well as semantic part, is introduced. This distance function is used to decide whether a group of pixels in an image forms a region. A number of image models formed from the combination of various approximation schemes and attribute sets are studied. These image models are tested on a number of real world examples, and a promising set of attributes and approximation schemes for region detection are reported. The results of region detection on a number of images using the proposed image models are presented.

Quadratic form schemes determined by Hermitian forms

Quadratic form schemes determined by Hermitian forms

An evaluation of eight discretization schemes for two‐dimensional convection‐diffusion equations

AbstractA comparative study of eight discretization schemes for the equations describing convection‐diffusion transport phenomena is presented. The (differencing) schemes considered are the conventional central, upwind and hybrid difference schemes,1,2 together with the quadratic upstream,3,4 quadratic upstream extended4 and quadratic upstream extended revised difference4 schemes. Also tested are the so called locally exact difference scheme5 and the power difference scheme.6 In multi‐dimensional problems errors arise from ‘false diffusion’ and function approximations. It is asserted that false diffusion is essentially a multi‐dimensional source of error. Hence errors associated with false diffusion may be investigated only via two‐ and three‐dimensional problems. The above schemes have been tested for both one‐ and two‐dimensional flows with sources, to distinguish between ‘discretization’ errors and ‘false diffusion’ errors.7 The one‐dimensional study is reported in Reference 7. For 2D flows, the quadratic upstream difference schemes are shown to be superior in accuracy to the others at all Peclet numbers, for the test cases considered. The stability of the schemes and their CPU time requirements are also discussed.

Accelerated convergence of the steepest descent method for magnetohydrodynamic equilibria

Iterative schemes based on the method of steepest descent have recently been used to obtain magnetohydrodynamic (MHD) equilibria. Such schemes generate asymptotic geometric vector sequences whose convergence rate can be improved through the use of the epsilon-algorithm. The application of this nonlinear recursive technique to stiff systems is discussed. In principle, the epsilon-algorithm is capable of yielding quadratic convergence and therefore represents an attractive alternative to other quadratic convergence schemes requiring Jacobian matrix inversion. Because the damped MHD equations have eigenvalues with negative real parts (in the neighborhood of a stable equilibrium), the epsilon-algorithm will generally be stable. Concern for residual monotonic sequences leads to consideration of alternative methods for implementing the algorithm.

Stability and accuracy of spatial approximations for wave equation tidal models

Amplitude and phase characteristics for numerical approximations to the shallow water wave equation are obtained for linear and quadratic finite elements, for finite difference approximations, for non-constant bathemetry, and for uneven node spacing. Stability is shown to require non-zero friction as well as satisfaction of a Courant constraint. Lumping is shown to reduce the Courant constraint for stability while higher order and quadratic finite element approximations require a more restrictive constraint than their second order and linear finite element counterparts. The amplitude of the propagation factor for stable schemes and propagating waves is seen to be independent of the Courant number and type of numerical approximation. Although the higher order and quadratic schemes provide better propagation of the low and moderate frequency waves, the highest frequency waves (2Δ x ) are better propagated by low order numerical methods.

Quadratic form schemes with non-trivial radical

Quadratic form schemes with non-trivial radical

Inversion of ABEL's integral equation — Application to plasma spectroscopy

This paper presents two approximate methods such as Quadratic and Cubic approximations for the Riemann-Liouville fractional integral and Caputo fractional derivatives. The approximations error estimates are also obtained. Numerical simulations for these approximation schemes are performed with the test examples from literature and obtained numerical results are also compared. To establish the application of the presented schemes, the problem of Abel’s inversion is considered. Numerical inversion of Abel’s equation is obtained using Quadratic and Cubic approximations of the Caputo derivative. Test examples from literature are considered to validate the effectiveness of the presented schemes. It is observed that the Quadratic and Cubic approximations schemes produce the convergence of orders h3 and h4 respectively.

A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers' equations

Solutions to the one- and two-dimensional Burgers' equations with moderate to severe internal and boundary gradients have been used to compare minimum truncation error three, five-, and seven-point finite difference schemes with linear, quadratic, and cubic rectangular finite element schemes. The various schemes demonstrate the theoretically predicted convergence rates if the mesh is sufficiently refined. In one dimension the quadratic finite element scheme and the five-point finite difference scheme are computationally the most efficient. In multidimensions the finite element method is less economical than the finite difference method even if the group representation is used for the convective terms. In two dimensions the linear group finite element representation and the three-point finite difference scheme are computationally the most efficient on a coarse mesh or if a severe gradient is present.

The calculation of some laminar flows using various discretisation schemes

For various laminar, two-dimensional flows, the upwind, hybrid and the quadratic finite-difference schemes are evaluated. The upwind and hybrid schemes are used in their normal forms. The quadratic formulation (Leonard's QUICK scheme) has in the past been found to be less stable than these two other schemes because it can, on occasion, generate both negative and positive influence coefficients. Two new forms of the quadratic formulation, both of which always generate positive influence coefficients, are presented for a uniform grid distribution and tested by reference to the results obtained from the upwind and hybrid schemes and data from other sources. These two new schemes are found to be stable, although one form seems to be sensitive to the size of the finite-difference grid. Furthermore, the solutions obtained using these new formulations are equal to or better than those obtained using the upwind and/or the hybrid schemes. It is concluded that these new formulations require further validation; and to this end, these new formulations are provided in a form that is easy to install into existing computer codes.

Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.

Quadratic finite element schemes for two‐dimensional convective‐transport problems

Quadratic finite element schemes for two‐dimensional convective‐transport problems