Conditional stability of explicit schemes in finite differences complicates the choice of a time step. The increase in the number of the grid nodes for more precise computations and the corresponding space step decrease leads to the increase in computer costs due to the decreasing of the time step. We...

The self-consistent evolution of the ion-acoustic collisionless shock with shock-reflected ions is numerically studied with kinetic simulations. We want to compare different approaches to the shock simulation based on the PIC-method. The results of one-dimensional fully kinetic simulation for both plasma components with the real ion-to-electron mass ratio are compared...

Today the method for the development of portable numerical simulation programs is very important because of the two main reasons. The first reason is the diversity of supercomputer architectures in Top500 and the second one is a demand for using the most powerful computers to simulate, for example, plasma. The...

The paper reviews the problem of forecasting the possible maximum pressure at the well-head, at the well-bore and at near-wellbore zone of reservoir during the process of new stimulation technology like reactive chemistry application. The technology provides stimulation by thermobaric effects. This impact occurs as a result of thermal decomposition...

This paper investigates a magma flow in permeable zones of the lithosphere. The model equations describing the dynamics of saturated porous media both without tangential stresses and with them are presented. The phenomenological method used for deriving the equations provides the thermodynamic correctness. Based on the model of the heterophase...

This paper deals with an ill-posed problem to determine the size distribution for water drops in a cloud from a given scattering phase function. Numerical experiments have shown that a method based on non-negative least squares with additional requirements of smoothness can be used to solve the ill-posed problems.

In order to create systems of vibroseismic monitoring for earthquakeprone areas it is needed to carry out the simulation of seismic wave propagation in the media typical of volcanic structures. To this end it is required to develop supercomputer technologies for decreasing the computation time and simulation of "big" 3D...

New parallel algorithm is developed for simulating the dynamics of a thin circumstellar disk. It is based on the combination of a gridless method of smoothed particle hydrodynamics (SPH) and the grid-based convolution method for calculating a gravitational potential. To develop the algorithm, we made software profiling of numerical experiments...

This paper is aimed at studying the efficiency of the electromagnetic radiation generation in various nonlinear processes occurring during the beamplasma interaction. The beam and plasma parameters were chosen close to the parameters in the experiment on the GOL-3 facility (BINP SB RAS). The research was conducted by means of...

In order to simplify the development of high-performance plasma physics codes for hybrid supercomputers, a template implementation of the Particlein-Cell (PIC) method was created. The template parameters are the problemspecific implementations of "particle" and "cell" (as C++ classes).

Thus, it is possible to develop a PIC code for the supercollider...

In this paper, a new hydrodynamic numerical simulation of interacting galaxies is proposed. The main subgrid physics processes are: the star formation, the supernovae feedback, the cooling function and the molecular hydrogen formation. The collisionless hydrodynamic approach was used for the simulation of the stars and dark matter. An approach...

The problem of network reliability calculation is studied. It is assumed that a network has unreliable communication links and perfectly reliable nodes. The diameter-constrained reliability for such a network is defined as probability that every pair of terminals of network is connected by operational paths with a number of included...

The atmospheric chemistry dynamics with a convection-diffusion model is studied. The numerical Data Assimilation (DA) algorithm presented is based on the additive-averaged splitting schemes. It carries out "fine-grained" variational data assimilation at separate splitting stages with respect to the spatial dimensions, i.e., the same measurement data are assimilated with different...

The main point of the performance of a code for a GPU (Graphical Processing Unit) is data locality. For the PIC method this means that all the particles belonging to one cell must be located closely in memory. During the particle push the particles might move to other cells, and...

We compare two algorithms for solving the Poisson equation: the first one is based on domain decomposition with direct coupling of subdomains (DDCS) and the second one is based on multidimensional Fast Fourier Transform and data transposition (FFTT).

Results of the comparison made helped us to introduce several optimizations for...

We present a new parallel algorithm for supercomputer simulation of gas-dust circumstellar gravitating disks. The algorithm uses the domain decomposition technique and combines numerical methods of smooth particle hydrodynamics (SPH), particle-in-cell (PIC) and grid-based gravitational solver with the convolution method and parallel multidimensional Fast Fourier Transform.

The algorithm is designed...

We present a 2D hybrid numerical plasma model for the simulation of the physical processes in supernova remnant shock precursor. In simulation, a shock is generated by sending a supersonic ow against a reflecting wall. The interaction between the incoming and reflected streams produces a sharp discontinuity, which moves away...

The problem of an electron beam and plasma interaction is considered. The physical mechanism of the beam-plasma interaction includes a resonant excitation of plasma oscillations, the occurrence of the plasma density modulation, followed by electron scattering. For the modeling, the PIC-method is used. In order to solve this problem, a...

The problem of simulation of plasma electron dynamics in the magnetic trap with inverse magnetic mirrors and multipole magnetic walls is considered. The model is built on the basis of Particle-In-Cell method. The complexity of processes under study and the necessary in a high precision of results required the development...

We present a parallel 3D algorithm for simulation of beam-beam effects in supercolliders, where colliding beams have superhigh densities and high relativistic factors. The algorithm is based on particle and domain decomposition and demonstrates good speed-up and scalability.

In this paper, a new scalable hydrodynamic code GPUPEGAS (GPU-accelerated Performance Gas Astrophysical Simulation) for the simulation of interacting galaxies is proposed. This code is based on a combination of the Godunov method as well as on the original implementation of the FlIC method, specially adapted to the GPU-implementation. Fast...

The main disadvantage of explicit schemes for the numerical solution to nonstationary problems is in a very strong stability condition for the size of a time step size. One of the possibilities to improve the efficiency of explicit algorithms is to use different time steps in different space subdomains. From...

A parallel version of the program for the simulation of flows in the Earth's mantle has been developed. A non-stationary model of the mantle flows describes a compressible medium with strongly varying rheological and transport properties. It is based on the solution of the Navier–Stokes equations. The numerical model includes...

In this paper, a modification of the high-tech expert system GIS-EEDB (Expert Earthquakes Database) intended for solving a wide range of seismological research tasks, called "Fukushima-EEDB" is proposed. For the first time the system was developed on the platform of Windows 8. The basic logical and functional structure as well...

A 3D kinetic study of the plasma relaxation processes caused by the propagation of an electron beam in high-temperature plasma was carried out. The mathematical model is built on the basis of the Particle-in-Cell (PIC) method. The performance for supercomputers powered by both Intel Xeon processors and Nvidia Tesla GPUs...

We present a new parallel algorithm for solving the Poisson equation in the context of non-stationary stellar dynamics problems, e.g. rotating galaxies or circumstellar disks. This allows us to conduct numerical experiments on a mesh with 10–100 billion of nodes and to use more than 10 thousand of processors. This...

This paper deals with the analysis of the hybrid plasma model based on the kinetic description of an ion component of the plasma and hydrodynamic approach for electrons. This type of models is widely used to investigate the processes and mechanisms of the collisionless interaction of interpenetrating plasma flows with...

The numerical 3D modeling of the gas-dynamic reactants flows of a flow reactor and their mixing that was made with the FLUENT software package is discussed. The presence of the modes where a reaction zone with a high content of C2 hydrocarbons is localized in the center of reactor presented....

A new technology for the simulation of physico-chemical processes in a reactive medium is proposed, which allows optimizing and adjusting kinetic schemes of chemical reactions. To implement the technology, a ChemPAK software tool, which is used to solve the primal problems of chemical kinetics, was developed. The proposed technological solution...

The Trust-Region (TR) algorithms are relatively new iterative algorithms for solving nonlinear optimization problems. High efficiency of the TR methods was demonstrated in a number of recent publications. They have the global convergence and local super-convergence, which differ them from the line search methods, commonly used for solving Inverse Problems....

We present numerical studies on the ion acceleration from thin foils irradiated by ultra-high contrast laser pulses. Two-dimensional particle-in-cell (PIC) simulations reveal that ions are for a certain time accelerated in a phase-stable way by the laser radiation pressure. The development of the Rayleigh–Taylor instability leads to destroy foils and...

This paper considers the problem of obtaining invariant solutions under rotation transformation in the expansion of a gas sphere into vacuum in the Cartesian coordinates for the two-dimensional case. Numerical results are presented and a comparative analysis of conventional and invariant difference schemes is made. Various non-invariant difference schemes of...

The paper deals with Monte Carlo simulation of the lidar return signals. Descriptions of mathematical model, simulation algorithms, and mathematical software are presented as well as results of several numerical experiments for ground-based and space-borne lidars.

A 3D kinetic study of relaxation processes caused by the electron beam propagation in high-temperature plasma was carried out. This problem has two different spatial scales: the plasma Debye length and the beam-plasma interaction wavelength, that is, some 10 or 100 times larger, thus one needs high-performance computing to observe...

We present the new method for computation of a gravitational potential for isolated systems in cylindrical coordinates. This method solves the main difficulty arising when treating isolated systems: in order to correctly state the Dirichlet problem for the Poisson equation at the boundary of a finite computational domain, one must...

The Particle-in-Cell methods (PIC) are widely used in the numerical simulation. The media under study in these methods are represented with a sufficiently large number of model particles with definite characteristics such as mass, charge, velocity. The evolution of a system of particles at each time step is evaluated in...

This paper describes a numerical algorithm and results of numerical calculations of the mass problem solution of determination of belonging of a set of points to a set of arbitrary figures covering an area. Such figures could be irrelative crossed or not. The problem is solved by the earlier described...

The problem of an electron beam and plasma interaction, arising from the GOL-3 (BINP SB RAS) experiments, is considered. For the given problem, it is appropriate to use the collisionless plasma approach, described by a set of the Vlasov–Maxwell equations. The Vlasov equation is calculated via the particle-in-cell method. To...

A graphical user interface of the program package ERA-DD for the input and editing of initial data for a two-dimensional boundary value problem is described. The boundary of calculation domain is approximated by pieces of straight lines and circle arches. The input of the information about the boundary is provided...

In this paper a new algorithm to reduce the noise effects in the Particle-in-Cell method for the Vlasov–Poisson system is proposed. The method is demonstrated on an example of a one-dimensional Riemann problem of plasma physics. Collisionless completely ionized non-isothermic plasma is considered. The model calculates only the ion movement...

The paper deals with a family of nonlinear gradient filters that can be applied to one noisy image or several quasi-identical patterns (i.e., several images of the same object with independent noise). Such filters are based on a specific wavelet decomposition and a preliminary statistical analysis of quasi-identical patterns. Results...

In this paper, several numerical characteristics of sample randomness are discussed. These characteristics are based on the concepts of waiting time and information level of samples. The considered numerical characteristics can be used for developing the novel methods for model selection in statistical analysis.

A technique for evaluation of temperature in the Particle-in-Cell method is proposed. A number of computational experiments were conducted in order to reveal how a grid step affects the temperature evaluation according to the proposed technique both in 1D and 3D cases for electrostatic plasma. In order to prove the...

We have constructed a quasi-stationary distribution function of the rotating collisionless gravitating disk with an oblate spheroid-like structure at the center. The construction technique is based on the numerical tracking of the evolution of the initially non-stationary thin disk on time scales of dozens of rotations. We show that a...

In this paper, the PIC method with an adaptive mass is introduced. This enables us to conserve the velocity distribution, and not only density, momentum, the center of mass, energy in a cell. This is important for simulating multi-stream flows. That is why this modification was tested on the Riemann...

In this paper, a numerical model for the propagation of the shear Alfven waves (SAWs) on open magnetic field lines using a hybrid kinetic approach is presented.

There is a two-dimensional hybrid model: an ion component of plasma is described by a standard set of the MHD equations, while electrons...

Formulas of monitor metrics are introduced for generating the vector field-aligned and/or adaptive grids. Some results of numerical experiments are demonstrated.

One algorithm to solve the 3D mixed boundary value problem for the Navier-Stokes system of equations is presented in this paper. The 3D mixed finite volume exponential type approximations on staggered grids are used. To solve the resulting system, an algorithm based on a three-level iterative method is proposed. Results...

The finite volume methods and technologies for the solution of the 3D elliptic boundary value problems (BVPs), with a complex geometry, on the quasistrucured grids are proposed. The grid data structure and the element-by-element approach for computing the local balance and assembling the global matrices are considered. The results of...

On the finite volume solution of the 1D parabolic nonlinear equation

This paper presents some numerical experiments with iterative solvers of algebraic linear systems for mixed finite element approximations.

We propose a special monotone reconstruction algorithm in which the monotone property of a high order scheme is improved with the help of the monotone first order scheme. The algorithm provides spreading the shock only at one grid point and ensures second order of the integral convergence through the shock...

This paper discusses the key-properties from the Parallel Finite Element Method (PFEM). It focuses at the PFEM applications in the context of non-conforming finite element basis functions (for maximal parallelism) on locally-bisection-refined tensor-product grids (for simple and cheap load balance techniques).

The Parallel Finite Element Method is an iterative solution...

This work is devoted to the construction and study of a computational scheme based on the mixed vector finite element method for modeling of the three-dimensional nonstationary electromagnetic fields. Numerical study of convergence of the mixed vector finite element method in the three-dimensional case on a class of problems, having...

Finite element methods (the Uzawa algorithm and a mixed finite element method) for the solution of the Navier-Stokes equations on triangular grids are considered. For approximation of velocity and pressure, interpolating functions from different finite element spaces are chosen. The properties of the algorithms are tested on an analytical test...

The implementation of the biconjugate and the squared conjugate gradient (BiCG and CGS) preconditioned iterative methods are described for solving non-symmetric systems of linear algebraic equations (SLAEs) which arise when approximating multi-dimensional boundary value problems (BVPs) for diffusion-convection partial differential equations (PDEs), by finite eifference, finite element and finite volume...

The main objective of the paper is to present a program to solve the 3D BVP for the problem of linear thermoelasticity. Numerical algorithms for data structures, element-by-element finite volume approximations, and iterative solution are given.

The purpose of the present program is to solve a mixed boundary value problem...

We consider algorithms of numerical solution of a nonlinear system of three diffusion-convection partial differential equations (PDEs).

The paper deals with a numerical model based on the finite element discretization of the 3-D thermoelasticity problem in compound parallelepipedal domain. The piece-wise trilinear functions are used. Iterative process is based on the Neumann-Dirichlet domain decomposition procedure, and numerical experiments demonstrate that the convergence rate does not depend on...

This paper considers the software of the integer and the mixed-integer quadratic programming, which is based on the method of branches and boundaries with one-sided branching. Some examples of the solution of test problems are presented.

Modern problems of the mathematical modeling include a very wide range of computational tasks. Those tasks are based on solving different problems of mathematical physics, especially, in engineering. Such systems can be considered as passing of sets of data through the nodes of a graph. So, our task is to...

The generator of algorithms to calculate a set of Vandermonde and Hahkel algebraic structures elements is proposed.

The paper presents an algorithm for the numerical solution of the initial value problems for systems of ordinary differential equations with singular matrix multiplying the derivative. The algorithm uses the (*m*,*k*) scheme of the Rosenbrock type with time-lagging derivative matrices, and the adaptive step size control...

This paper is dealt with investigation of the numerical aspects concerned with using the vector finite element method for solving non-stationary electromagnetic problems. A special variational formulation and its discrete analogues are offered. Peculiarities of inputting a source current into such statements are considered. The results of some numerical experiments...

In this work we investigate some features of numerical implementation of the vector finite element method of lower orders for different types of elements. The comparison of data structures, computer memory requirements and application of iterative solvers for nodal and vector finite element approximations are presented.

An ILU(0) modification for sparse matrices storage technique called the CSlR is discussed and briefly analyzed. Details of program implementation for the GMRES(m) preconditioning are described for C language.

We present here general principles of organization of the module "Resonance" for stochastic simulation, which is a part of the package "ACCORD", and describe basic procedures of the module for simulation of random variables.

The on-line library ACCORD includes algorithms for solving different tasks of mathematical modeling. The description of the algorithms is in the special database, which contains description of algorithms, the author software source code and executable components, which allows the users to try algorithms. The access to the library resources is...

The main topic of the paper is to present a program package to solve 3D BVP for the Helmholtz-type equation. Numerical algorithms concerning data structures, approximations, and solving are described. Structure of the input data file and the usage examples are presented. Some useful recommendations for users are given.

Two explicit incomplete factorization methods and their program implementation are presented for solution of linear algebraic systems with real symmetric positive definite (SPD) matrices. The algorithms are based on the efficient Eisenstat modification of preconditioning for the matrix row sparse format. The fast iterative convergence is provided by conjugate gradient...

The generator of *M*-stage modified RK-scheme, 2 < *M* < 21, for solving the ODE *y*' = *f*(*t*,*y*) with initial conditions and the Volterra integral equation (by the RK-method of advanced accuracy) is proposed. Modified and classical RK-methods differ for a triangular matrix *B* only are used.

The algorithm and the code for computing several eigenvalues and their corresponding eigenvectors of a large sparse symmetric positive definite (SPD) matrix, which arises as a result of grid approximations (FDM, FEM, or FVM) of multidimensional boundary value problems (BVPs) are described. The preconditioned inverse iteration (PINVIT) method is implemented...

The paper presents an algorithm for the numerical solution of the initial value problems for implicit systems of ordinary differential equations (ODE). The algorithm uses the Rosenbrock-type scheme with time-lagging derivative matrices, and the adaptive step size control for the global error. Some examples of solution of test problems are...

The paper introduces an algorithm for the numerical solution of initial value problems for systems of ordinary differential equations (ODE). The algorithm uses the Rosenbrock-type and the Runge-Kutta-type schemes with Jacobian freezing and automatic step size control policy based on the global error estimation. Some examples of solution of test...

The paper presents the software for the linear programming problems both for the integer and the mixed-integer linear programming. For the solution of problems of the above types it is possible to exploit a parallel algorithm. Some examples of solution of test problems are given.

In the paper, we continue the investigation carried out earlier, namely, another approach to design two level explicit schemes for solution of the boundary value parabolic problems is proposed.

We consider the three-dimensional Dirichlet problem for the equation
Δ*u* + (*v*, grad *u*) + *cu* = -*g*, *u*|Γ = *ψ*
in a domain Ω with the boundary Γ, which is assumed simply connected and piecewise smooth. We suppose the functions *v*, *c*, and...

We consider a system of spatially homogeneous Smoluchowski equations. We will suggest that the coagulation coefficients are finite. Among numerical methods for solving this problem, Monte Carlo algorithms, based on the direct simulation of the coagulation process in a model particle system, play an important role. Note, that the Smoluchowski...

We consider the problem of estimating the reflected light intensity. This problem arises when studying the interference of backward scattering in laser sensing of the ocean from the atmosphere. Let *tα* be the time it takes for the intensity to achieve some asymptotical function. In this paper, a new...

One of advantages of the Monte Carlo methods consists in capability to evaluate various functionals by weighting estimates, for instance, it is possible to evaluate the eigenvalues by using the estimate of the parametric derivations of the solution. The other significant application of weight estimates consists in capability of parametric...

The high accuracy numerical solution of the Laplace or the Poisson equation is reduced to a sequence of the more simple finite difference problems of the second order accuracy. Some algorithms to realize effectively this scheme are discussed.

We consider an algorithm for the stochastic simulation of the Gaussian three-dimensional fields with a discrete argument and with regard the dependence of horizontal correlation functions on the vertical coordinate. The area for uses for the algorithm in question for a specific class of correlation functions of the horizontal fields...

The paper is devoted to the substantiation of the direct numerical method for the integral equation of the first kind with logarithmic singularity on the closed curve; it is based on the piecewise-linear approximation of unknown function from its values at quasiuniform grid and collocation condition in the same gridpoints....

Convection-diffusion problems are models for describing the transportation of matters in a diffusive medium. They are also a part of more general equations, such as the Navier-Stokes equations appearing in fluid flow problems. We consider the linear convection-diffusion problem of the special form and show that under the corresponding conditions...

In this paper, a probabilistic approach to solving some inverse and direct problems to the telegraph equation is presented. The multidimensional cases and specific features of the inverse problems, where it is commonly required to determine only the functional of solution, make the application of Monte Carlo method reasonable.

It is well-known that Monte Carlo methods are used for solving various urgent problems in mathematical physics. A lot of new numerical stochastic algorithms and models were elaborated in Institute of Computational Mathematics and Mathematical Geophysics (ICM and MG) SD RAS during last years. The Monte Carlo schemes can be...

The problem we discuss here is how to construct the local smooth *VP*-splines preserving the monotonicity of data from the point of view of the regions of their parameters. Also, we will give a practical recommendation how to satisfy these conditions if data and the slopes are given and fixed.

The problem of the construction of "good" grids for the numerical solution of multidimensional boundary value problems (BVPs) in complicated computational domains has a big history and extensive special literature.

There are two main conventional requirements for the discretization of BVPs. The first one consists in the approximation quality which...

In this paper, we study the convergence of the fictitious domain method for solving a system of grid equations for the finite element method that approximates the third boundary problem for the differential equation
Δ2*u* + *au* = *f*
in the piecewise bicubic Hermit interpolations subspace of *W...*

Using the diagonal transfer method, a partial solution of a linear system of equations is found. The description of corresponding algorithm is given. Computational costs of the algorithm are discussed.

The classical problem of the transformation of the orthogonality weights of polynomials by theory of the space *Rn* is discussed. The described system of polynomials - pseudo-orthogonal on the discrete set of *n* points - is a new result. The polynomials of this system, as the orthogonal ones, are...

In this paper, we are interested in computing the smallest eigenvalue and its corresponding eigenvector of a large symmetric positive definite matrix.

The theorems on traces of functions from the Sobolev spaces play an important role in studying boundary value problems of mathematical physics. These theorems are commonly used for a priori estimates of the stability with respect to boundary conditions. The trace theorems play also very important role to construct and...

A new version of the direct Monte Carlo method for solving boundary value problems for the Boltzmann equation is presented. In contrast to the conventional approach, we do not solve the problem via stabilization in time; when evaluating functionals of the solution to the Boltzmann equation, the random trajectories are...

This paper completes the analysis of the Maxwell operator with impedance boundary conditions for arbitrary time dependence.

The paper deals with the new classification of the ranges in image based on simple geometrical consideration. This approach leads to the fast algorithms for the training of fractal bases and fast coding-decoding processes in the image compression. Some numerical examples are also presented.

The aim of this paper is to introduce some formal procedures for the optimization of parameters in energy functional for variational spline interpolation. For the abstract splines with the tension we obtain the representation formula which shows the structure of dependence of spline with the tension with respect to tension...

We consider an interpolation problem. We have the given values
*fi* = *f*(*xi*)
of some periodic function *f*(*x*)
of the period *b - a* in the nodes of a mesh

Δ: *a* = *x*0 < *x*1 < ... < *x*N = *b*.

It is required to construct a (*b - a*)-periodic interpolating spline *S*(*x*) of the degree 2*n* - 1 (*N* ≥ 2*n*).

The paper deals with the "true" multi-dimensional interpolation problem at scattered meshes with a huge number of interpolating points. For its solution we suggest here a new numerical technology consisting in partitioning of the problem on a number of subproblems and in a successive glueing of solutions to the subproblems....

In this paper the approximations of diffusion-convection equation, describing the charge transfer in semiconductors are concidered.

The paper deals with studying of some preconditioning operators providing unconditional convergence of difference schemes for solving parabolic problems. Three examples of such operators are considered. There are a preconditioner of the domain decomposition method of the Neumann-Dirichlet type and two preconditioners of the fictitious domain methods for the Neumann...

In the article, we propose and study one modification of the finite element splitting algorithm for the solution to the Neumann boundary parabolic problem with mixed derivatives. We consider the simplest equations with constant coefficients without advective terms. In our case, for the Neumann problem, the error estimate contains the...

Recently an algebraic multilevel incomplete factorization method for solving large linear systems with the Stieltjes matrices has been proposed. This method is a combination of two well-known techniques: algebraic multilevel (AMLI) and incomplete factorization. However, the efficiency of this method strongly depends on the choice of the relaxation parameter *θ...*

In this paper we describe the program of triangular mesh construction with Delaunay properties for the domains, which boundaries consist of straight lines and arcs of circles and ellipses. There was used the geometric preprocessor TTNS in order to form the discret similarity for the calculated domain and to construct...

The analog of the implicit Runge-Kutta method applied to Volterra integral equations of the first kind is considered. It allows to obtain the results of high accuracy under a sufficient simplicity and stability of used algorithm. The estimation of numerical results for a fixed time step is performed. A special...

The paper considers the algorithm of numerical solution of the Focker-Plank-Kolmogorov equation for the probability density of a solution of a stochastic differential equation. Its solution is approximated by cubic splines on the time-dependent moving grid.

One of the most effective approach in solution of mesh and finite-element SLAEs *Au = f*, arrising in approximation of two-dimensional (or multi-dimensional) problems is the decomposition method. The essense of the method consists in special choice of easy-invertible linear transformation *H* and in a successive realization of iterating process
*u...*

In the article we propose and study a new noniterative domain decomposition algorithm without overlapping subdomains and with the use of splitting procedure in one of subdomains for solution multidimensional boundary value parabolic problems.

In the article we propose and study a noniterative domain decomposition algorithm without overlapping subdomains and with the use of the penalty functionals on the interface between subdomains. Such type of algorithms were consideredearlier. In all these works the error estimate for optimal penalty parameter is *O = √τ*. In this...

The main topic of the paper is to present a way of constructing the second order finite-volume approximations on nonuniform grids to solve 3D mixed boundary value problems for diffusion equation with piecewise constant coefficients. For obtaining the difference equations, a linear combination of the balance relations for the normal...

A strongly *S*-stable (by A. Prothero and A. Robinson) one-step noniterated method is presented. Results of numerical calculation showing the advantage of the proposed method in comparison with a similar *L*-stable method are given.

In this paper we consider an abstract spline smoothing problem in Hilbert space and express Newton’s iteration formula for an optimal choice of the smoothing parameter α in terms of the residual operator
*Rαz = z – Aσα*.

An effective multistep algorithm for numerical solution of Volterra integral equations of the second kind, based on the implicit Runge-Kutta (RK) method, is constructed. The choice of the Gauss scheme for the implicit RK method allows to obtain algorithm, having a higher approximation order for one-step method and maintaining the...

In this paper we will deal with the approximate solution of Fredholm's and Volterra's equations with the kernel of the kind *K*(*x-t*). We shall use the known algorithm for the search of the approximate solution in the form of a linear combination of preassigned basic functions*φ...*

The aim of this paper is to suggest optimization procedure for the refinement of energy functional in variational spline approximation problem. Our approach is based on a separation of measurement data between two sets. First is the set of basic measurements (nodes of spline), second is the set of control...

In this paper we investigate the asymptotic mean-square stability (m.s.-stability) of the family of numerical methods for solving SDE's in the Ito-sense generalizing Rosenbrock's type methods. The connection between the asymptotic m.s.-stability of the numerical method for solving SDE and the absolute stability of the corresponding Rosenbrock's type method are...

The paper considers the questions of the numerical analysis of stochastic auto-oscillating systems. The low computer costs variable stepsize algorithms was constructed for solving the non-linear stochastic differential equations. There are given results of numerical experiments obtained with the help of the dialogue system "Dynamics and Control".

To solve the tree-dimensional static problems in elasticity theory in the displacements a new class of effective iteration methods – factorized operator-triangular methods was studied. The additive expansion of the diagonal operator leads to the analogous expansion of the initial matrix operator in the sum of triangular-matrix operators. The degree...

By means of the fictitous regions method locally two-sided estimations the initial boundary value problem with the second order parabolic equation of the first type are obtained.

The paper deals with studying the domain decomposition algorithm on two subdomains, where for one of them, which contains sufficiently small number of nodes, is used explicit scheme with small time step, and for another subdomain may be used effective direct algorithm (for example, subdomain is parallelepiped). This method is...

A class of nonlinear 1-D parabolic equations with known solutions are introduced. The computer programs for estimation of the absolute errors of numerical methods are decribed.

Strong convergence of interpolating splines on the *imbedded meshes* is established without the assumption that the system of operators corresponding to the added interpolation conditions is *correct*. It is also shown that correctness of the system of operators is equivalent to the zero intersection of their kernels.

The necessary...

Letter to Editorial Board on the book of A.Yu. Bezhaev and V.A. Vasilenko **"Variational spline theory"**.

Wave process in a one-dimensional vertically-inhomogeneous medium induced by a sounding impulse moving from the depth is considered. Mathematical background of the algorithm for the reconstruction of the medium's mechanical parameters is given when the form of the initial wave and the surface seismogram are known. Theoretical results are illustrated...

The paper deals with some results, which are connected with the numerical solution of elasticity problems in the case of an arbitrary curvilinear coordinates system. The main ideas will be illustrated for the case of the Cartesian coordinates system.

In the paper new concepts of construction and analysis of numerical schemes of solving ODE systems are used for RK-method of advance accuracy. The analysis of compatibility of equations, the expression for the truncation error are given. The new analitic technics is based on the combinatorial identities. The constructed theory...

In this paper the recurrent relation generating the chain fraction is considered. We find different non-weakened sufficient conditions of its boundedness and small growth of computation errors, caused by the inaccuracy of arithmetical operations implementation. These conditions as well as error estimates do not contain any indefinite constants and are...

The mixed spline approximation problem combines the peculiarities of the problems of spline interpolation and smoothing that were studied by many mathematicians. The monograph of Loran should be mentioned specially. It gives the conditions of existence and uniqueness of the interpolating and smoothing splines in the general form. In practice...

The paper presents the method of constructing the difference analogs of elliptic operators based on the use of their factorized structure (for second order equations as an example). For the Poisson equations the structure of the difference operators obtained allows us to suggest a new efficient method for solving the...

Let us correct the mistake in paper Lotova G. **Calculation of time constant of particle breeding by Monte Carlo method using parametric derivatives** // Numerical Analysis. — 1993. — # 4. — P. 27-34.

In this paper two nonsolved problems of the Monte Carlo theory are presented. The first of them concernes the uniform boundedness of the "walk on spheres" estimates for the Helmgoltz equation. Another problem is the important example from the minimax Monte Carlo theory for evaluating of many integrals.

In this paper the algorithms of Monte Carlo methods for solving the *n*-dimensional Helmholtz equation are investigated. The dependence of the computational efficiency of the algorithms on *n* is studied.

Convergence of randomized spectral models of homogeneous vector fields is studied in the sense of convergence in distribution in a uniform metric of the Banach space of continuous functions. Under quite moderate restrictions on the parameters of the spectral model, weak convergence to a Gaussian field is shown if the...

The paper contains the results of time constant calculations for the process of particle breeding. The calculations are based on the estimates of parametric derivatives of the particle flux. The transfer process of radiation is assumed to be stationary.

**Attention!** Please, also see this article.

In this paper some methods of statistical simulation of non-Gaussian non-stationary scalar and vectorial processes and non-homogeneous spatial fields with continuous argument on the basis of synthesis of discrete models and models on point fluxes are considered.

Numerical models of vector-valued random fields are extensively used in solving applied problems. These models have become the subject of many investigations. The paper deals with methods of numerical modeling of homogeneous vector-valued random fields based on the spectral decomposition. General relations for spectral models are obtained and particular algorithms...

New probabilistic representations for systems of elliptic equations are constructed in the form of expectations over the Markov chains. It is shown that this approach gives the effective Monte Carlo algorithms even in the cases, where the classical probabilistic representation based on the Wiener and diffusion processes fails. As an...

Numerical stochastic procedures for estimating integrals depending on parameter are considered. The discrete mesh on the domain of definition of parameter is introduced, and the Monte Carlo algorithms for estimating integral in mesh points are used. The independent Monte Carlo estimates and the "depended tests" method are compared. It is...

New numerical method for approximating two-dimensional flow field for the viscous incompressible fluid in the vicinity of the flat boundary is introduced. Using the vorticity formulation of the Prandtl equation we come to the heat equation with nonlinear right-hand side. We consider various boundary value problems for this equation and...

The article is devoted to the new Monte Carlo method for the calculation of covariance function of the solution of biharmonic equation when its right-hand side is a random field. The comparison of this method with the randomization algorithm of the Monte Carlo method is presented. The numerical results of...

Preface

Introduction: A Guide to the Reader

2. Reproducing Mappings and Characterization of Splines

3. General Convergence Techniques and Error Estimates

5. Interpolating Dm-splines

7. Vector Splines

8. Tensor and Blending Splines

9. Optimal Approximation of Linear Operators

10. Classification of Spline Objects

11. ΣΠ – Approximation and Data Compression

12. Algorithms for Optimal Smoothing Parameter

Appendix 1. Theorems from Functional Analysis Used in this Book

Appendix 2. On Software Investigations in Splines

Literature

Index

This paper deals with the problem of mean-square stability of numerical methods for solving SDE's. We introduce the notion of the stiff in a mean-square sense system of SDE's, the practical verification of which is not difficult. In the capacity of the investigated family of numerical methods the generalization of...

The paper contains short description of ΣΠ-algorithm for the approximation of the function with two independent variables by the sum of products of one-dimensional functions. Some realizations of this algorithm based on the continuous and discrete splines are presented here. Few examples concerning with compression in the solving of approximation...

The goal of this paper is investigation of the efficiency of iterative preconditioned cojugate gradient methods for solving linear systems of equations which arise in *p*-*h*-version of finite element methods. The results of numerical experiments are presented for the model boundary value problem with different values of...

Non-stationary problems of finite element solutions require an efficient generation of sequential meshes with minor changes in density functions of points distribution. The Delaunay meshes sequential generation is studied in terms of successive insertions and removings of points following changes in density function.

Modern Runge-Kutta method of solving ODE bears a slight resemblance with the classical (explicit) method and is based on the transformation of the differential equation to the integral one. The contents of the mathematical theory was formulated by J.C. Butcher et all. Nevertheless, the technique of constructing fundamental equations of RK-method...

The transfer of charge carries in a semiconductor device for stationary conditions is described by elliptic differential equations with oscilating coefficients. The uniform convergence of one-dimensional Scharfetter's scheme on the whole interval is shown in this paper.

The paper deals with studying the domain decomposition algorithm on two subdomains, where for one of them, which contains sufficiently small number of nodes, is used explicit scheme with small time step, and for another subdomain may be used effective direct algorithm (for example, subdomain is parallelepiped). This method is...

The abstract variational theory of splines in the Hilbert space originated from the well-known paper by M. Atteia (1965) and supported by P.J. Laurent's researches (1968, 1973) is today a well-developed field in the approximation theory. We mean that the forthcoming researches in abstract theory were initiated by the problem of high-quality...

The problems of determining the structure of numerical method, the choice of its parameters, analysis of meansquare or weak convergence of the numerical solution to the true one are much more complicated for systems of SDE, than for those of ODE. Nevertheless, many theoretical and practical ideas of the numerical...

Modern Runge-Kutta method of solving ODE bears a slight resemblance with the classical (explicit) method and is based on the transformation of the differential equation to the integral one. The contents of the mathematical theory was formulated by J.C. Butcher et all. Nevertheless, the technique of constracting fundamental equations of RK-method...

The paper deals with studying the domain decomposition algorithm with overlapping subdomains. This algorithm is based on the splitting method and uses the additive presentation of some bilinear form. Earlier this method was described for two subdomains. In our consideration we formulate the decomposition algorithm for an arbitrary number of...

The purpose of this paper is to construct the interpolating function as a ratio of two splines. The numerator and the denominator of this ratio minimize some combined variational functional on the set of pairs of functions which satisfy interpolating conditions and some additional restrictions. Such a construction was proposed...

In this paper we give an algorithm for finding the bounds for a ratio of two neighbouring mesh steps which provide the convergence of odd-degree spline interpolants and their derivatives. For the quintic splines numerical values are obtained which improve the estimates by S. Friedland, C. Micchelli.

The uniform error estimates with respect to a small parameter are obtained here for the finite element approximation of the elliptic boundary value problem with a small parameter. The space of trial functions is the space of special *L*-splines with the basis of local functions.