Numerical Methods of Mathematical Analysis

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This lead appeared in the Institute of Mathematics in the laboratory (later department) headed by Yu. S. Zavyalov which was organized within the Department of Computing Engineering in the beginning of the 60’s. Investigations that were carried on in the Laboratory were aimed at mathematical support of developments in the field of creation of computational media and systems, including mathematical modeling of physical processes appearing in the technology of computing machinery elements production. In the beginning of the 70’s these themes were concentrated in the Laboratory of Numerical Methods of Models Investigation of the Department and further developed in parallel with methods of spline functions.

Mathematical modeling of film electromechanics.

In modeling of such devices as a film electrostatic relay whose operation is based on hysteresis, the nonlinearity manifests itself in the branching of solutions that correspond to boundary value problems describing the static equilibrium state of elastic elements depending on a parameter. In the case of a relay, the potential difference applied to movable and stationary electrodes plays the role of the parameter which is responsible for solution branching. Therefore, the computational side of the problem consisted in creating methods of investigation of nonlinear problems that would take into account the possibility of appearance of multiplicity of solutions in the process of numerical construction of the solution-to-parameter dependence. It should be noted that to find the parameter values at which there occurs branching of solution, as well as to construct all solutions (both stable and unstable), was of main practical interest, since important operational characteristics of a device (in the case of a relay, the voltages of switching and release) were computed as a result.

The basics of the film electromechanics as a new scientific lead that emerged in the Institute of Mathematics in the laboratory headed by V. L. Dyatlov are presented in the monograph:

V. L. Dyatlov, V. V. Konjashkin, B. S. Potapov, S. I. Fadeev, Film Electromechanics. Novosibirsk, Nauka, 1991. 248 p.

Besides the results of investigation of electromechanical converters of energy, in this monograph one can find detailed description of methods of numerical investigation of nonlinear boundary value problems for systems of ordinary differential equations in general statement and computation results for concrete models of film electromechanics. At present time, the ideas of film electromechanics are widely used in development of micromotors which will find application in various fields of engineering, medicine, etc.

Mathematical modeling of catalytic processes.

In the beginning of the 70’s, co-operation with the Boreskov Institute of Catalysis in the field of mathematical modeling of catalytic processes on catalyst particle and in reactors of various types began to form. It turned out that from computational viewpoint nonlinear boundary value problems describing steady-state regimes have the same nature as mathematical models of film electromechanics. Numerical analysis is aimed here at finding critical values of parameters, in particular, solution branching points, in whose neighbourhood sharp changes in proceeding of stationary regimes are observed. Thus, results of computation allow to obtain, in the frames of the mathematical model, the knowledge of the margins of changes of parameters, within which the process is stable.

The end of the 80’s is marked by close co-operation with the department headed by M. G. Slin’ko (Karpov Institute of Physical Chemistry, Moscow) in connection with development of software package AWP-CT (automated working place of a chemist-technologist) in the Department. Original algorithms of numerical investigation of mathematical models of an ideal mixing reactor and processes on catalyst particle that were created in the Laboratory were included into the package. By present time, over 20 joint works dealing with problems of mathematical modeling in catalysis were published in collected volumes, central and foreign journals. Researchers of the Laboratory participated and continue to participate in several joint projects on RFBR grants, international grants, and integration projects of SB RAS.

It should be noted that in development of mathematical models of microcatalysis allowing to explain experimental data, important role is played by variations in statements of boundary value problems connected with selection of the process kinetics, with accounting of some or other balance conditions, etc.; and the “flow” of boundary value problems arising here requires to conduct a numerical express analysis. The same can be said about models of film electromechanics. As a reaction to the situation emerged, an effective version of the method of solution continuation by a parameter for a nonlinear boundary value problem in a sufficiently general statement was proposed by the middle of the 80’s in the Laboratory. Here we mean the boundary value problem for a system of ordinary differential equations of first order with two-point nonlinear boundary conditions connecting the values of unknown functions at the endpoints of the interval.

The advantages of the method were described in detail in the above-mentioned monograph. As a short description it can be noted that in the proposed method a discrete model is set into correspondence to the nonlinear boundary value problem, this model being obtained by application of the method of spline collocation on a nonuniform mesh with approximation error of 4th order. In this case the approximate solution is sought in the form of Hermite cubic splines of class C1. To find the solution to the system, a version of the method of continuation by parameter is used, that version allowing to effectively construct the solution-to-parameter dependence with account of possible appearance of solution branching of “rotation” type. Continuation by parameter is accompanied by adaptation of the mesh, which plays important role in appearance of solutions with large gradients whose position is not known in advance. Lastly, the “one-stepness” of the scheme allows to include into the family of considered boundary value problems the problems in which the right-hand sides of the system of differential equations have discontinuoties of first kind in the independent argument.

Later the above method of continuation by parameter was carrieed out as a software package BPR-Q for personal computers and also packages of the same type for (a) boundary value problems for combined systems of differential and finite (algebraic) equations, (b) auto-oscillations, and (c) emitted nonlinear oscillations, including oscillations under action of a rectangular pulse.

The same ideas of the method of continuation by parameter were used for numerical investigation of systems of finite nonlinear equations, in particular, for systems describing stationary solutions to autonomous systems, including determination of stability of stationary solutions by the Godunov-Bulgakov method (package STEP). The effectiveness of the proposed computing means was demonstrated by the present time on many practical problems. This development won recognition in many institutions of RAS and abroad. The development of this theme owes much to the support of Academician S. K. Godunov.

Application of developments in teaching process.

Finally, one should mention application of the above-mentioned developments in teaching process as a constituent part of the mathematical support of corresponding computing practical works (see the educational publication Software package STEP for numerical investigation of systems of nonlinear equations and autonomous systems of general type. Description of operation of the package STEP on examples from the course “Engineering chemistry of catalytic processes”. Authors: S. I. Fadeev, S. A. Pokrovskaja, A. Yu. Berezin, I. A. Gajnova).

The experience of numerical investigation of nonlinear boundary value problems, both accumulated in the Laboratory and known from literature, shows to the necessity to develop other variants of the method of continuation by parameter. For example, the method based on differential sweeps in combination with the procedure, which was proposed earlier, of selection of the “current” parameter for continuation of the solution by one step. Another non-formalizable and therefore interesting problem is connected with various methods of parametrization of the problem with the aim of further application of the method of continuation by parameter. Besides, further efforts are necessary in improvement of computational means (software packages) to make them more convenient for the user. Evidently, the last task implies close co-operation with developers of mathematical models.

Mathematical modeling by applications of spline functions.