Classical Numerical Method Research Articles

Modeling the Relationship between Water Uptake by Cucumber and NaCl Accumulation in a Closed Hydroponic System

In a completely closed hydroponic system, Na and Cl commonly accumulate in the root zone, at rates depending on the concentration of NaCl in the irrigation water (rate of Na and Cl inlet) and the Na to water and Cl to water ratios at which they are taken up by the plants (rates of Na and Cl outlet). However, while the concentration of NaCl in the irrigation water is commonly a constant, the Na to water and Cl to water uptake ratios are variables depending on the concentrations of Na and Cl in the root zone and, hence, on the rates of their accumulation. To quantify this feed-back relationship, a differential equation was established, relating the rate of Na (or Cl) accumulation to the rate of water uptake. This equation was solved according to the classical Runge-Kutta numerical method using data originating from a cucumber experiment, which was conducted in a fully automated, closed-loop hydroponic installation. Four different NaCl concentrations in the irrigation water, 0.8, 5, 10 and 15 mm, were applied as experimental treatments. The theoretically calculated curves followed a convex pattern, with an initially rapid increase of the Na and Cl concentrations in the root zone and a gradual leveling out as the cumulative water consumption was rising. This was ascribed to the gradual approaching of the Na to water and Cl to water outlet ratios via plant uptake, which were increasing as NaCl was accumulating in the root zone, to the constant NaCl to water inlet ratio (NaCl concentration in irrigation water). The model could predict the measured Na and Cl concentrations in the drainage water more accurately at 10 and 15 mm NaCl than at 0.8 and 5 mm NaCl in the irrigation water. Possible explanations for these differences are discussed. Plant growth and water uptake were restricted as salinity was increasing, following a reverse pattern to that of Na and Cl accumulation in the root zone. The leaf K, Mg and P concentrations were markedly restricted by the increasing salinity, while that of Ca was less severely affected.

A new model for prediction of dispersoid precipitation in aluminium alloys containing zirconium and scandium

A model has been developed to predict precipitation of ternary Al 3(Sc, Zr) dispersoids in aluminium alloys containing zirconium and scandium. The model is based on the classical numerical method of Kampmann and Wagner, extended to predict precipitation of a ternary phase. The model has been applied to the precipitation of dispersoids in scandium containing AA7050. The dispersoid precipitation kinetics and number density are predicted to be sensitive to the scandium concentration, whilst the dispersoid radius is not. The dispersoids are predicted to enrich in zirconium during precipitation. Coarsening has been investigated in detail and it has been predicted that a steady-state size distribution is only reached once coarsening is well advanced. The addition of scandium is predicted to eliminate the dispersoid free zones observed in scandium free 7050, greatly increasing recrystallization resistance.

Variance reduction for Monte Carlo simulation of stochastic environmental models

To determine the probability of exceedence Monte Carlo simulation of stochastic models is often used. Mathematically this requires the evaluation of an expectation of some function of a solution of a stochastic model. This can be reformulated as a Kolmogorov final value problem. It can thus be calculated numerically by either solving a deterministic partial differential equation (Kolmogorov's Backwards equations) or by simulating a large number of trajectories of the stochastic differential equation. Here we discuss a composite method of variance reduced Monte Carlo simulation. The variance reduction is obtained by the Girsanov transformation to modify the stochastic model by a correction term that is obtained from an approximate solution of the partial differential equation computed by a classical numerical method. The composite method is more efficient than either the standard Monte Carlo or the classical numerical method. The approach is applied to estimate the probability of exceedence in a model for biochemical-oxygen demand.

A research on grey numerical imitation and modeling of groundwater seepage system

Based on grey set, grey numbers and their operation properties, the grey numerical model of groundwater seepage system was set up for the first time, the whole grey solving method of the model was given and it was proved that the common solving method of the model was only a special case of the grey solving methods. At the same time, the grey solving method was compared widely with common solving method, classical numerical method. The study shows that the grey solving method is better in depicting the procedure of transporting grey data of groundwater system. On the basis of the theoretical study, two basic kinds of cases about groundwater seepage were selected: the prediction of pit yield and the evaluation of groundwater resources on a groundwater basin. In the cases, systematical analyses were made for generalization and greylization of the hydrogeologic conditions, setting up of the grey model, identification and correction of the model as well as its prediction and evaluation. It was pointed out that when the grey numerical model is used to predict pit yield, the upper limit of the “grey band” of groundwater level cannot be higher than planed safe groundwater level, when evaluating the groundwater resource, the lower limit of the “grey band” of groundwater level cannot be lower than controlled level of groundwater.

Fast valuation of financial derivatives

A method for pricing European contingent claims (options) based on Monte Carlo simulation with variance reduction is presented. The evolution of the option price can be formulated as a Kolmogorov final value problem and thus be calculated numerically either by solving the deterministic partial differential equation or by simulating a large number of trajectories of the corresponding stochastic differential equation. The authors discuss a Monte Carlo simulation method combined with variance reduction obtained from a Girsanov transformation of the stochastic differential equation by a correction term that is obtained as a rough solution of the partial differential equation computed by a classical numerical method. The trade-off between these methods is investigated and it is shown that the composite method is more efficient than either the standard Monte Carlo or the classical numerical method.

A computational method for solving quasilinear singular perturbation problems

A class of quasilinear, singularly-perturbed, two-point, boundary value problems for second-order, ordinary differential equations without interior turning points is considered. To solve these problems Newton's method of quasilinearization is adopted. Then the resultant linear problems are solved by the numerical method suggested in [1]. The method presented in [1] is a combination of an exponentially-fitted finite difference method and a classical numerical method. Further, it is based on the boundary value technique [2] generally used to solve singularly-perturbed boundary value problems. Error estimates for the numerical solution of linear problems are stated. Some examples are given to illustrate the method.

Analysis of the FK5 System in the Equatorial Zone

15 catalogues produced in the eighties and 12 catalogues of the sixties-seventies have been used to assess the consistency of the FK5 system with observations in the declination zone from −30° to 30°. Positions of the FK4-based catalogues have been transformed at the equinox and equator J2000.0. Classical δ-dependent and α-dependent systematic differences (Cat-FK5) have been formed for individual instrumental systems of the catalogues by a method close to the classical Numerical Method. The weighted mean instrumental systems for the two subsets of catalogues centered at the epochs 1970 (MIS 60–70) and 1987 (MIS 80) and for all types of systematic differences have been constructed. The mean errors of the total systematic differences in α and δ have been estimated as 14 mas and 21 mas, respectively, for the catalogues of the 60-70ies, and 10 mas in both α and δ for the catalogues of the 80ies.It has been found that the mutual consistency of individual instrumental systems of catalogues of the 80ies with respect to δ-dependent systematic differences is superior by the factor 1.5 comparing with the catalogues of 60-70ies, while the consistency of both catalogue selections with respect to α-dependent systematic differences is comparable. Random accuracy of the FK5 positions and proper motions at the epochs under analysis has been assessed as close to expected from the formal considerations. Actual systematic discrepancies of the FK5 with observations at the respective epochs have been detected. For systematic differences δαδ cosδ and δδδ, the absolute deviations of the MIS 80 are, in general, within 40 mas, those of the MIS 60-70 are within 30 mas. For systematic differences δαα cosδ and δδα, the absolute deviations reach 30-40 mas for both MIS. For total systematic differences, local deformations of the FK5 system in the equatorial zone in both right ascension and declination has been found exceeding expected ones from the formal errors of the FK5 system by a factor about 1.5 for the MIS 60-70, and by a factor about 2 for the MIS 80. Consistency in area distribution between both MIS for the total systematic differences δαcosδ has been detected. Quick degradation of the FK5 system with time due to optimistic estimation of the errors of its proper motion system is supposed to be one of the main causes of its discrepancies with observations. The results in declination are recognized to be less reliable due to larger inconsistency of the individual instrumental systems.Before the next space astrometric mission will be realized, ground-based observations will continue to be the only available check of an external systematic accuracy of the HIPPARCOS catalogue. Evidently, random and, possibly, systematic accuracy of each individual catalogue observed from the Earth surface would be inferior to that of the HIPPARCOS catalogue. Taken as an ensemble, however, a certain selection of catalogues might give a rather definite idea about the actual distortions of the HIPPARCOS system. This study shows to which level of accuracy such ensembles of different selections of catalogues might check the HIPPARCOS system in the equatorial zone. The analysis of the FK5 gives also an idea about levels of random and systematic discrepancies which may be expected in the equatorial zone when the HIPPARCOS catalogue will be compared with the FK5 at different epochs.

Semi-automatic computer construction of three-dimensional shapes for the finite element method

Precise estimation of spatio-temporal distribution of ions (or other constitutives) in three-dimensional geometrical configuration plays a major role in biology. Since a direct experimental information regarding the free intracellular Ca 2+ spatio-temporal distribution is not available to date, mathematical models have been developed. Most of the existing models are based on the classical numerical method of finite-difference (FD). Using this method one is limited when dealing with complicated geometry, general boundary conditions and variable or non-linear material properties. These difficulties are easily solved when the finite-element-method (FEM) is employed. The first step in the implementation of the FEM procedure is the mesh generation which is the single most tedious, time consuming task and vulnerable to mistake. In order to overcome these limitations we developed a new interface called AUTOMESH. This tool is used as a preprocessor program which generates two- and three-dimensional meshes for some known and often-used shapes in neurobiology. AUTOMESH creates an appropriate mesh by using the mesh generator commercial tool of FIDAP.

Auditory nonlinearities: A new approach in light of an active model

A new model of ear which simulates the nonlinear behavior of the peripheral auditory system for real input sounds is described. The main assumption underlying the structure of the model is the presence of an active mechanism which works mainly near the resonance. The general structure of the model looks like classical lumped transmission line. Each cell has its own frequency resonance according to the coordinate‐frequency correspondence observed along the basilar membrane. A nonlinear local feedback from the resistive element is mixed to the input signal in each cell. Such a circuit exhibits interesting results concerning, mainly, the evolution of selectivity with respect to the input signal level, two‐tone suppression effects, and combination tones generation. The first experiments carried out by a computer simulation which uses a classical numerical method for resolution of a nonlinear differential equation system are presented. Tuning curves which show a decreasing Q factor when increasing the input level, and the results obtained with multifrequency component input signals, are presented.

A semi-numerical method for computing the second-order perturbation of artificial satellites

In this paper a new semi-analytical, semi-numerical method for computing the second order perturbation of artificial satellites is presented. The basic ideas are the following: 1. 1. We adopt as fundamental elements σ ∗ , which are the mean elements obtained after subtracting the short-period terms Δσ s from the osculating elements. 2. 2. The secular and long-period rates (up to third order) of σ ∗ are found by numerically averaging the rates of the osculating elements. 3. 3. The osculating elements in computing dσ/ dt are found by adding to σ ∗ the short-period terms Δσ s , in which the first-order terms are obtained by analytical method, and the second-order terms by Fourier analysis. By using σ ∗ , we have overcome the shortcoming in the numerical method, due to the use of short integration steps, while our numerical computation of dσ ∗/dt and Δσ s (2) obviates the development of complicated formulae. Our computing time of the second-order perturbation is only 1 10 that of the classical numerical method, and our required memory is about 1 5 of the purely analytical method. In the numerical method, an improved Chebychev iteration process is developed to better serve the needs of satellite dynamical geodesy. Detailed formulae for computing the perturbating forces and some numerical results of computation by our method are given. These show that the radius and across-track errors are less than 0.1m, and that the along-track errors are less than 1m.