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Abstracts of papers

Madrane, A., El Boukili, A., Vaillancourt, R.
A new overlapping unstructured grid algorithm.

The objective of this paper is to propose an efficient and robust algorithm to automatically localize the intermesh boundaries for the overset unstructured hybrid grid method. The intermesh boundaries are localized using the minimum wall distance concept proposed by Nakahashi et al. and the advancing front vicinity algorithm for the hole-cutting procedure. In such a case, since the elements are not deformed during the computation, the costly re-meshing part of conventional methods is prevented. This method relies on the inter-grid boundary definition to establish communication among independent grids in the overset system. At the end the Navier-Stokes equations are integrated on several overset systems to examine the capabilities of this method.
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Pelletier E.F., Vaillancourt R.
Modelling instrument's sounds using Malvar wavelets.

An instrument's sound is the result of a collection of overtone frequencies (harmonics) that occur simultaneously when a note is played. This collection is responsible for the 'colour' of the sound or the 'timbre' of an instrument. A music note consists of three main parts, the attack, sustain and decay, which describe the beginning, middle and end of the sound-each of these has some positive duration, perhaps very short. We propose a mathematical model that takes into account an instrument's harmonic frequency features during the attack, sustain and decay like a synthesizer which attempts to imitate the sound of a particular instrument by replicating the amplitudes of its harmonics, a method called 'additive synthesis'.
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Hua X-H., Vaillancourt R.
Dynamics of permutable meromorphic functions.

Unshifted and shifted multiscaling functions are used as mathematical models for curve fitting of irregularly sampled data. This pre-processing procedure, combined with multiwavelet neural networks for data-adaptive curve fitting, is shown to perform well in the case of high resolution. In the case of low resolution, it is more accurate than numerical integration and cheaper than matrix inversion. The curve fitting method for irregularly sampled data can be applied to a pre-processing design for the discrete multiwavelet transform.
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Sharp P.W., Vaillancourt R.
Error growth of some symplectic explicit Runge-Kutta Nyström methods for a simulation of the gas giants.

At one extreme of large stepsizes, the global error for symplectic explicit Runge-Kutta Nyström (SERKN) methods applied to a long simulation of the four gas giants, Jupiter, Saturn, Uranus and Neptune, consists entirely of truncation error and typically grows as time t. At the other extreme of very small stepsizes, the global error of SERKN methods for the same simulation consists entirely of round-off error and grows as t2. We use numerical testing to investigate how the global error grows for stepsizes between these two extremes.
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Sharp P.W., Vaillancourt R.
Efficient order-five second-derivative explicit Runge-Kutta pairs with interpolants.

It is well known that six stages are needed to form an order-five explicit Runge-Kutta (ERK) formula, if all the stages are evaluations of the first derivative f. There are two ways this order barrier might be overcome. One is to using stage values from the previous step, giving rise to two-step ERK methods. The other is to use evaluations of second and higher derivatives. Ono {J. of Information Processing, , No. 3 (1989), 251-260] showed that if one of the stages was an evaluation of f' at the start of the step, it was possible to get an order five formula using four evaluations of f and one evaluation of f'. These new methods are particularly appealing for many problems where f' is easy to work out and inexpensive to evaluate. The usefulness of Ono's methods can be extended by making them variable-stepsize and by appending an interpolant. To this end, we present two families of pairs, consisting of formulae of orders four and five and construct the pairs in such a way that an order-five, globally C2 interpolant can be formed without using extra evaluations. Order five interpolants for conventional ERK pairs require eight evaluations and are typically just C1 globally. Thus the new interpolants offer a significant advantage when the interpolant must be evaluated frequently.
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Kolyshkin A., Nazarovs S.
Calculation of the coefficients of the Ginzburg-Landau equation for shallow water flows.

Numerical method for the calculation of the coefficents of the complex Ginzburg-Landau equation for shallow water flows is presented in the present paper. Linear stability problem is solved by a pseudospectral collocation method based on Chebyshev polynomials. Critical values of the parameters are calculated by means of IMSL routine DGVCCG. Explicit formulas for the coefficients of the Ginzburg-Landau equation are presented. The coefficients are calculated for different values of the parameters of the problem.
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Iltins I.,
A.Temkin's method of separation of variables.


There is non-stationary temperature field in infinite long, empty cylinder is acquired by using A.Temkin's method of variables separation. Obtained expression of temperature field calculation is significantly simpler than the analogue expression following from Fourier's method of variables separation. A.Temkin's method of variables separation is applied for calculating temperature field in an interval of time where temperature field is determined only by boundary conditions and is not affected by initial conditions.
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Kolyshkin A., Vaillancourt, R., Volodko I.
Approximate method for the calculation of the change in impedance due to a flaw in a conducting cylindrical layer.

Two approximate methods for the calculation of the change in impedance of a cylindrical conducting tube are considered in the paper. The methods considered are the layer approximation and the Born approximation. The results of numerical computations of the change in impedance by means of the two methods are compared with experimental data. It is found that the layer approximation does not work well for cylindrical conducting layers.
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Jekabson G., Lavendel J., Evaluation of model selection criterions in multiple nonlinear regression analysis.

A new exact analytical formula for the impedance change used in non-destructive testing problems is derived. The derivation is based on the Green's formula in contrast with the previous studies that used Lorentz theorem for obtaining the formula known in literature. The new formula for the impedance change has the form of a triple integral of scalar product of two vector potentials: the vector potential in the flaw and the vector potential in the same region in the absence of the flaw over the region containing the flaw. The similar formula obtained earlier by previous authors has the form of a triple integral of scalar product of amplitude electric field vectors. It is strictly proved that the new simple formula is equivalent to the previous formula used in literature.
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G. Burov.
The Cyclic Structure of an Associative Matrix.

A method that can help to improve the accuracy of computing algorithm for matrix inversion is offered. The goal was to develop an algorithm for calculation of permanent of square matrix with the help of address positional structures in additional space. They are formed in cyclic form without the use of recurrent methods. Due to such approach accuracy of operation of matrix inversion can be increased. These structures are formed from local addresses of information allocation in an associative matrix. The analysis of numerical algorithms for matrix inversion is carried out on the basis of combinatory configurations in abstract information space. Mathematical connections between parameters of algorithm are described using index expressions. The new approach of their formation in the cyclic form of address lexicographic combinatory configurations (ALCCs) is developed Algorithmically correct ALCCs for calculation of permanent of an inverse associative matrix are developed as well. For this purpose ALCCs are formed from ordered numerical sequences. Such sequences are mapped into classes, which are oriented relative to the components of a positional vector of an associative matrix. ALCCs are formed using combinatory functionals and classifications of its fragments. For these purposes it is offered to use operators with combinatory properties. Properties of such operators are considered and their parallel structure is proved. Return to Contents

G. Burov.
Combinatory models of inversion of special type matrices
In this article the problem connected to the increase of accuracy of operation of the inversion of badly conditioned matrices of a special type is considered. The algorithm of the inversion has been developed on the basis of principles of the combinatory analysis. It is offered to use address lexicographic combinatory configurations (ALCCs), which are formed on the basis of index matrix grids of associative matrices in such algorithms. The method of ALCC formation using the information-compressed lexicographic models of their description is developed. Algorithmically correct ALCCs for calculation of permanents of the inverse associative matrix are developed. The method of association of identical ALCC fragments in one address group is offered. Positional structures of ALCC of associative matrices are formed on hierarchical principles as a decomposition of independent blocks. This allows to create economic algorithms with parallel properties for solving systems of equations. The majority of operations of information processing is carried out in the space of ALCC using the methods of monitoring. These methods are based on information interchange between numerical information space and ALCC space. Efficiency of the offered approach has been checked on a numerical example of the inversion of badly conditioned 20th order Hilbert matrix. Results have been received with 100% accuracy. Their credibility has been verified with the help of a special technique based on the use of ALCC, which allows to solve the problem of information processing programmatically with big accuracy for cases when the length of numbers exceeds the number precision limit of the computer.
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Spalvins A., Slangens J., Janbickis R., Lace I., Juhna T.
The hydrogeological model of the Baltezers, Rembergi and Zakumuiza water supply complex.

The regional spatial hydrogeological model has been created for the Baltezers, Rembergi and Zakumuiza waterworks. Methodology of forming the model is described, especially, in part of simulation of infiltration flow. The model will be used for optimising regimes of the waterworks and also for predicting after effects
of contaminant migration endangering quality of groundwater supplied by the waterworks.
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Spalviņs A., Slangens J., Janbickis R., Lace I., Juhna T.
Hydrogeological model for simulation of contaminant migration for the area of the 3rd and 10th infiltration pools of the Baltezers waterworks, Latvia

The local hydrogeological model has been created to simulate contaminant migration for the area of 3rd and 10th infiltration pools of the Baltezers waterworks. Migration of algae toxins from the pools for the worst case scenarious has been investigated by applying the MODPATH and MT3D systems. Due to dilution and bio degradation processes, the concentration of toxins in production wells is at least ten times lower than in the pools. Therefore, even in the case of short migration time and slow biodegradation, considerable decrease of the impact of the toxins may be expected.
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Spalvins A., Slangens J., Janbickis R., Lace I., Skibelis V., Eglite, I., Macans, A.
Improved hydrogeological model for evaluating contaminant migration in groundwater polluted by sulphur - sludge pools at Incukalns, Latvia.

In 1998, a hydrogeological model for the contaminated Incukalns place has been created and rough prognoses of SO4 and surface active components migration in groundwater have been obtained. In 2004-2005, this model has been considerably improved in order to find the best methods of stopping pollution plumes and of lessening their impact on local environment. The improved model is described and new results obtained are presented.
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