Low-order Schemes Research Articles

Anharmonic oscillator as a test of the coupled-cluster method

Recent applications of the coupled-cluster method to the (${\ensuremath{\varphi}}^{4}$${)}_{2}$ quantum field theory showed the necessity of a detailed analysis of its convergence behavior. The anharmonic oscillator has always served as a good test case for different approximation schemes. We treat the model both in the maximum-overlap condition and in the Hartree approximation. The ground-state energy is reproduced very well for all values of the coupling strength and already for a low-order truncation scheme. The expansion in correlation amplitudes can be carried out in extremely high order and shows a divergent tendency of the amplitudes. Introducing temperature dependence allows us to select the stable ground state out of a variety of solutions of the hierarchy of equations.

CONTROL STUDIES IN AN EXTRACTIVE DISTILLATION PROCESS: SIMULATION AND MEASUREMENT STRUCTURE

This work explores the use of variable measurement and control structures in the state estimation and control of distillation processes. Distillation processes with complex behavior are difficult to control (azeotropic and extractive separation) due to the existence of sharp temperature gradients and were found to be ideal candidates for the implementation of variable control structures. An on-line sequential analysis of the temperature measurements along the length of the column allows a fast selection of the best measurements to be used in low order, simple and robust control schemes. Overall results indicate that variations in column operating conditions could also cause variations in the resulting control schemes. For the case of extractive distillation, more robust estimation and control were obtained through the use of the proposed methodology.

Analysis of some low-order finite element schemes for the navier-stokes equations

Various low-order finite element schemes for the Navier-Stokes equations are analysed, including the commonly used bilinear velocity, piecewise constant pressure element, and the algorithms based on the finite element analogue of the MAC stencil discussed in a companion paper. By performing a truncation error analysis using a variety of different assumptions about how the nodal parameters represent the data, it is shown how a more complete picture of the behaviour of the algorithms can be obtained. In particular, differences in performance on coarse grids using algorithms which have the same asymptotic convergence rate can be explained.

Experiments with some low-order finite element schemes for the navier-stokes equations

It is shown how a consistent finite element approximation to the Navier-Stokes equations can be constructed on a general grid using an arrangement equivalent to the MAC stencil. It is known that the standard Galerkin construction does not give a consistent result unless at least quadratic velocities are used in this type of element. However, the use of either a finite difference approximation to the pressure gradient term or a Galerkin scheme equivalent to a vorticity method appears to give a consistent scheme. Both alternatives are tested, and the vorticity method is shown to be superior.

A numerical study of vortex shedding from rectangles

The purpose of this paper is to present numerical solutions for two-dimensional time-dependent flow about rectangles in infinite domains. The numerical method utilizes third-order upwind differencing for convection and a Leith type of temporal differencing. An attempted use of a lower-order scheme and its inadequacies are also described. The Reynolds-number regime investigated is from 100 to 2800. Other parameters that are varied are upstream velocity profile, angle of attack, and rectangle dimensions. The initiation and subsequent development of the vortex-shedding phenomenon is investigated. Passive marker particles provide an exceptional visualization of the evolution of the vortices both during and after they are shed. The properties of these vortices are found to be strongly dependent on Reynolds number, as are lift, drag, and Strouhal number. Computed Strouhal numbers compare well with those obtained from a wind-tunnel test for Reynolds numbers below 1000.

Perturbation theory of the electron correlation effects for atomic and molecular properties. Second- and third-order correlation corrections to molecular dipole moments and polarizabilities

The many-body perturbation theory is applied for the calculation of the second- and third-order correlation corrections to the SCF HF dipole moments and polarizabilities of FH, H2O, NH3, and CH4. All calculations are performed by using the finite-field perturbation approach. The pertinent correlation corrections follow from the numerical differentiation of the second- and third-order field-dependent correlation energies. This computational scheme corresponds to a completely self-consistent treatment of the perturbation effects. The third-order corrected dipole moments are in excellent agreement with the experimental data and the best results of other authors. A comparison of the present perturbation corrections for polarizabilities with the PNO–CI and CEPA results of Werner and Meyer reveals that some cancellation of the third- and fourth-order correlation contributions can be expected. The second-order corrected polarizabilities are as a rule better than the results of the third-order perturbation approach. It is concluded that also for polarizabilities the low-order many-body perturbation scheme is able to account for the major portion of the relevant correlation effects.

Variable measurement structures for the control of a tubular reactor

The feasibility of the variable measurements structure is investigated within the framework of controlling a tubular chemical reactor. An on-line sequential analysis of the temperature measurements along the length of the reactor, allows a fast and time-adaptive selection of the best measurements to be used in a low order, simple and robust control scheme. The approach has been developed for both static and dynamic cases. It is based on a recursive computation of the state estimate errors. Its theoretical and computational simplicity makes it attractive for on-line implementation. The procedure has been applied for the adaptive tubular reactor under pseudo-steady state and dynamic conditions with very encouraging results.

Design of low-order control schemes using reduction technique

Abstract A new design procedure is presented to obtain a simplified control scheme using the reduction technique. In some practical cases, the optimal control may be required using only a restricted set of state variables. An attempt is made to design a reduced-order control law using only a restricted sot of state variables. The technique is illustrated by an example.

Implicit Differencing of Predictive Equations of the Boundary Layer

Abstract The Crank-Nicholson method may not give useful results in detailed prediction of the thermal planetary boundary layer unless tune steps on the order of 10 s are used. In similar problems, lower order time differencing methods give reasonable results with time steps as large as 300 s. The reason for the superior behavior of the lower order schemes relative to straightforward application of the Crank-Nicholson technique is due to a better treatment of short waves which appear to be critically important in nonlinear terms.

Local errors of difference approximations to hyperbolic equations

It is shown that the rate of convergence of numerical approximations to the solution of hyperbolic partial differential equations is degraded when the solution being sought is not sufficiently smooth. A pth-order finite-difference scheme may give worse than pth-order convergence if the ( p + 1)st derivatives of the solution are not piecewise continuous. Spectral methods, which are normally expected to give infinite-order convergence, give only finite-order convergence if some derivative of the solution is not continuous. Near a discontinuity propagating along a characteristic of the differential equation, the truncation error of difference approximations is much larger on one side than on the other, and it oscillates in sign on the side where it is larger. (It may also oscillate on the other side of the discontinuity, depending on the order of the numerical scheme used.) Finally our analysis shows that, even if the true solution is not smooth, high-order schemes are more accurate than lower-order schemes.

A data structure for cognitive information retrieval

A new data structure developed in connection with a natural language question-answering system is described. The data structure is based upon a new high-order calculus. It allows a great degree of expressiveness and is suitable for extensive logical deduction. The inadequacies of low-order schemes for the representation of natural language information are given. A formal description of the new high-order structure which overcomes these inadequacies is then presented along with the properties that make it more suitable and attractive for machine processing of natural language information.