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A technique for calculating the problem is described and the results of calculations are given, where the need for a solution arose after unsuccessful attempts to take into account the experimental data within the frame-work of the Euler equations. The essentially new element in this method is the implicit scheme in Lagrangian variables with non-rectangular nets.

A two-dimensional system of the Stokes type, in which the coefficient of the highest order derivatives is a small parameter e >0, is considered, and a version of the multigrid method is developed to handle a finite-element scheme for the system, obtained by using biquadratic elements for the velocities, linear elements for the pressure and an approximatively-solenoidal basis. Approximatively-solenoidal interpolations are defined and used to design the coarse-grid bases. Numerical experiments demonstrate a high rate of convergence even for very small e values.

The three-dimensional dynamical problem of the oblique impact of a rigid pellet on a deformable elastoplastic barrier is solved using a hybrid grid-characteristic scheme for the numerical solution of non-stationary systems of hyperbolic equations. Thegrid-characteristic hybrid scheme is adapted for the numerical solution of multidimensional non-stationary problems in the mechanics of deformable bodies.

The two- and three-layer versions of a symmetric implicit difference scheme on distributed nets in the case of the one-dimensional equation of gas dynamics in the Euler form are considered. The approximate viscosity of the scheme is investigated by the method of differential approximation. The change in the approximation properties of the scheme when a mobile net is employed is analyzed. The results of a numerical experiment are presented.

A difference scheme is constructed for solving the initial value problem for one-dimensional Vlasov equations. The scheme is shown to be stable, and to converge to the solution of the continuous problem. It is used to solve a model problem concerning the charged-particle density distribution along the axis of a linear accelerator.

Using generally accepted methods of linearizing non-linear equations with subsequent local stability analysis, it is shown that a difference scheme for calculating the shift of a shock wave region, presented in a certain paper relating to the Godunov method, is unstable if considered in isolation. A simple scheme which has no such drawback is discussed here, and the problem of extending the algorithm to more variables is analyzed.

THE PAPER is primarily a survey concerned with method; it describes a scheme and technique for deriving and justifying equations in slow time (averaged equations) from the point of view of the stroboscopic method. Typical classes of problems drawn mainly from monographs, are discussed, in which the averaged equations are obtained by the classical Krylov-Bogolyubov procedure.

AN ALGORITHM monotonizing the solutions obtained by the MacCormack difference scheme is discussed. The noteworthy improvement in the accuracy of the description of gasdynamic discontinuities is demonstrated by the example of the simplest plane discontinuities of supersonic flows.

A recurrence scheme is proposed for classifying patterns when the input and output vectors are random. The asymptotic method is compared with Tikhonov's regularized method of least squares.

THE PROBLEM of finding a max-mim with respect to connected sets is considered with fairly general assumption. The problem is approximated by a sequence of simpler problems. In the light of a definition of approximation with respect to a functional, necessary and sufficient conditions for convergenece of the approximation are obtained. A min-max optimal control problem, illustrating the general scheme, is examined.

The set of trajectories isolated from the k-th 1.s.a. will be denoted by {Tk}. We will describe a trajectory by a series of numbers rP,kr,,A . . . rvjk=RlrA, where /J is the ordinal number of a trajectory isolated from the k-th l.s.a., j is the ordinal number of a fragment of the p-th trajectory, not including logical conditions and contained between logical conditions, r is the number of operators {AR} in this fragment.

Exact difference scheme operators for a system of equilibrium equations of a non-homogeneous anisotropic elastic body rigidly clamped in the rectangle Ω are used to construct difference schemes which have O(h2) accuracy in the grid norm of W21(ω) if the solution of the original problem belongs to the space W 2 3 (Ω) .

A scheme which enables one to construct converging iterative methods of minimizing a wide class of functions is proposed. The problem of minimizing an arbitrary locally Lipschitz function specified with errors is considered as an example.

FOR THE linear operator equation of the 1st kind Au = f, the convergence of the general scheme of finite-dimensional approximation (discretization) of the Tikhonov-regularized family of approximate solutions is examined, along with the application of this scheme to projection and quadrative methods.

The problem of the convergence of the conservative differential-difference scheme of gas dynamics in Euler coordinates is considered, taking account of the sources and sinks in the isothermal case.

A difference scheme for calculating high-speed non-stationary two-dimensional flows with regions of strong non-uniformity is proposed. The scheme is based on a divergent form of differential equations combined with methods of splitting and linearization of the nonlinearity.

A model of recognition algorithms for the initial information given in the form of arbitrary bounded domains in an n-dimensional Euclidean space Rn is constructed. The correctness of the algebraic closure of this model without constraints imposed on the final sample of control objects is proved, and the stability of the correct algorithm in the closure of finite power is investigated. A basic scheme for constructing the algorithms for recognition problems with a continual control set of objects converging to a certain degree to the correct algorithm is given.

Rothe's scheme for a non-linear equation of evolution with a singularity is considered. The convergence of the scheme is proved, and the error is estimated.

Some conservative schemes of the kinetic algorithm for solving gas-dynamic problems are examined. A two-layer difference scheme, closest to complete conservativeness, is constructed in Eulerian variables. A model two-dimensional problem of a point explosion is examined as a test.

A special difference net is constructed which is compressed in the boundary layer region of a singularly-perturbed equation with a turning point. It is proved that the simplest monotonie scheme on the resulting net converges regularly with respect to a parameter with first-order accuracy.

A variational-gradient method is described for solving operator equations. The method is shown to be convergent, and its rate of convergence is studied. A suitable computing scheme, realizable on a computer, is given, and the results of a numerical experiment are discussed.

MESH approximation of boundary value problems for differential inequalities is investigated, taking the example of an ordinary second-order differential inequality. The mesh inequalities are approximated by a non-linear difference scheme, which is solved by an iterative method. Convergence theorems and the results of numerical computations are givens.

FROM a family of three-layer completely conservative difference schemes approximating the one-dimensional equations of gas dynamics in Lagrangian mass variables, one scheme was successfully split into two systems of difference equations each of which, if the equations are considered in a definite sequence, can be solved explicitly. The results of trials of the scheme in the solution of two test problems are given.

A family of two-layer completely conservative difference schemes is constructed for the spatially one-dimensional equations of gas dynamics. It is possible to obtain complete conservativeness of the difference scheme by profiling the time weights in a space matched to the solution of the problem.

A new algorithm is given for calculating with high accuracy all the complex zeros of the modified Bessel function of the 2nd kind and its derivatives, when the argument takes complex values, and the index takes both real and complex values. The algorithm is based on new asymptotic formulate for the modified functions of the 1st and 2nd kind, and on a Newton iterative scheme. Curves show how the zeros move as the index varies, and there are tables of the zeros, calculated to nine significant digits.

A new difference scheme for the equations of the one-dimensional motion of a viscous heat-conducting gas in a magnetic field is studied. In the non-linear statement of the problem, the existence, uniqueness, and stability of the solution of the scheme are proved and error estimates are obtained in terms of the data. All the results are obtained in the large with respect to time.

An implicit finite-difference scheme of decomposition with respect to the spatial directions and physical processes is proposed using the generalised method of artificial compressibility. This scheme is suitable for calculating stationary flows of a viscous gas at arbitrary Mach numbers including values which may be as small as may be desired.

New algorithms are proposed for calculating with high accuracy the complex zeros of Hankel functions of the first and second kind when the index acquires real values and the argument acquires complex values. The algorithms are based on reducing the problem of calculating the zeros of Hankel functions to the problem — partly considered earlier by the authors — of calculating the zeros of a modified Bessel function of the second kind. Using asymptotic methods, analytical formulas are obtained which enable us to localize the zeros in the complex plane and find their first approximations, which are further refined using Newton's iteration scheme. A careful qualitative analysis of the zeros is made, programs∗∗ (∗∗One can consult the texts of all programs mentioned in this paper at the Computational Centre, Academy of Sciences of the USSR, Moscow.) for calculating both the Hankel functions and their derivatives themselves, and their zeros, are compiled, and a large number of zeros, some of which are presented in the paper together with graphs illustrating the regularity of the distribution of the zeros, are calculated.

Methods for computing modes with peaking in processes of heat conduction are justified. It is shown that, under certain conditions, the difference scheme for the equation of heat conduction with constant coefficients and a source gives the peaking time of the initial equation correctly. Extension of the method of proof to other equations is discussed, including quasilinear equations of parabolic type with a source.

New algorithms are proposed for calculating, with high accuracy, a Bessel function of the second kind Y v ( z ) and its derivatives, and also their complex zeros, when v takes real values, and z takes complex values The algorithms are based on the application of asymptotic methods to calculate the initial approximations of the zeros and on their further refinement using Newton's iteration scheme. A thorough qualitative investigation of the complex zeros is carried out, and new relationships in their behaviour are revealed. A large number of complex zeros, some of which are presented in the form of tables, is calculated.

A class of fourth-order variational elliptic inequalities with a boundary inside the region is considered. Using the penalty and net methods, the variational inequality is approximated by a nonlinear difference scheme. Estimates are given of the rate of convergence of the penalty method, and also an estimate of the rate of convergence of the solution of the difference scheme to the solution of the variational inequality.

A general scheme for constructing complete systems of functions in diffraction theory is described. The technique which has been developed enables one to obtain complete systems solely by specifying the geometry of the set of locations of the secondary sources.

A GENERAL scheme is described for obtaining a priori error estimates for the Bubnov-Galerkin method in arbitrary Banach spaces. As applications, the first and second boundary value problems for strongly elliptic systems of arbitrary order are considered. For these problems we obtain a priori error estimates for the Bubnov-Galerkin and Ritz methods, and for the variational-mesh method, in the norms of W5.

A difference scheme is proposed for two-dimensional flows of a compressible gas, comparable in efficiency with schemes for the equations of an incompressible fluid. The efficiency of the scheme is determined by an original iterative process, using a procedure for eliminating the pressure, which is characteristic for the equations of an incompressible fluid. A comparison with other schemes is made for a model convection problem.

Definitions of ω-stability and Liouville methods of numerical integration are introduced for systems of second-order ordinary differential equations. The existence of them properties guarantees that the difference scheme of some important qualitative characteristics of the initial dynamic system is preserved. The existence of these properties in Runge-Kutta type methods is analysed.

The effect of rounding errors when eliminating variables in sets of linear inequalities using Fourier's algorithm is analysed, and a calculational scheme which is stable with respect to the rounding errors is proposed. An estimate of the number of perturbations introduced by the rounding errors is given.

The first boundary value problem for a system of non-linear differential equations describing the strong bending of flexible thin plates is discussed.

A variational difference scheme on a rectangular mesh is considered, for studying non-linear problems of the propagation of light beams without axial symmetry.

The boundary value problem for the full potential equation is approximated by a conservative mesh-projection scheme of the method of integral identities. The additional dissipation needed to isolate a stable solution in supersonic velocity domains is introduced by modifying the expression for computing the density in accordance with the ideas of artificial compressibility. The resulting system of mesh equations is linearized at each iteration and is solved by a multimesh algorithm.

A method of constructing explicit difference schemes of second-order of accuracy on a minimal pattern is described. The method enables the required solution to be written uniquely in the class of piecewise continuous functions. As an example the two-dimensional non-stationary supersonic flow past an energy source is computed.