Tedious Iterations Research Articles

Air flow dynamics around an aerofoil by the stabilized finite difference method

A high performance computing scheme to visualize air flow dynamics around an aerofoil is developed, based on the finite difference method with stabilizing devices. A mass diffusion term is added to the equation of continuity to stabilize the finite difference equation. The time derivative of kinetic energy in the energy balance equation is assumed to be replaced by the time derivative of the pressure, to avoid the tedious iteration for the pressure term. The scheme is applied for two-dimensional compressible flow on an aerofoil in unsteady state. Numerical integration has been executed by a microcomputer with i486DX CPU at 40 MHz, on my compiler, for 200 × 500 mesh nodes, and the result is graphically displayed. The velocity vector pattern shows that the ground effect works to stretch the effective area of the aerofoil.

Accurate extraction of reaction probabilities in IR multiphoton dissociation by a focused Gaussian beam: an improved analytical method

An improved analytical method has been developed to extract the dissociation probability from the raw data in infrared multiphoton dissociation with a Gaussian-profile beam using a focused irradiation geometry. The conventional analytical method based on the power-law model is shown to overestimate the dissociation yield by as much as 60% compared with that predicted by the newly developed analytical method based on the cumulative log-normal distribution model. The obtained analytical solution is well fitted bysimple algebraic expressions, which enable us to analyze the experimental data quickly and accurately without tedious iteration procedures.

The estimates of surface heterogeneity by series expansions

In the foregoing treatment, Cerofolini's method for the estimate of the surface heterogeneity, based on the relation (1), has been analyzed from the standpoint of numerical mathematics. The limited amount of observed data, their experimental error and the properties of regressional procedures led to notable modifications of the original method. The main result is that the series expansion does not give straightforward solution of the problem, tedious iterations are necessary even for simple analytical forms off(U). Serious difficulties are introduced by the consideration of the incipient second-layer coverages, by existence of a number of maxima inf(U) and by existence of homogeneous regions. Deleterious impairment can be caused by differences in coefficients for the expansions of observed data (N, p) and ofθ(p, T, U). As in other methods, there is no possibility to estimate the absolute value of the least adsorption energyU0, only the relative quantitiesq are extracted from a single adsorption isotherm. In spite of the large amount of needed computations, the method does not give any possibility for estimates of the reliability of the calculated distribution. This contrasts with the recently developed method [8], in which the observed data are correlated by the sum (10). By means of this method, continuous and discontinuous distributions can be dealt with equally well and it can be easily tested whether any supposed distribution fits the data within experimental error. When isotherms at several temperatures and/or calorimetric data are at hand, analysis of the thermodynamic properties of different adsorption sites can be performed. The chief difficulty of the method, the solution of a system of non-linear equations, is much less tedious than the treatment analyzed in this paper.