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There was a new direction of studying of the complex systems last years, considering them as networks. Nodes in such networks represent elements of these complex systems, and links between nodes – interactions between elements. These researches deal with real systems, such as biological (metabolic networks of cells, functional networks of a brain, ecological systems), technical (the Internet, WWW, networks of the companies of cellular communication, power grids), social (networks of scientific cooperation, a network of movie actors, a network of acquaintances). It has appeared that these networks have more complex architecture, than classical random networks. In the offered review the basic concepts theory of complex networks are given, and the basic directions of studying of real networks structures are also briefly described.

This paper presents a universal method for constructing an agent-based model of complex systems for their further clear computer representation by means of object-oriented programming languages. The method specifies both steps of model developing from the mathematical description of the system to the determined architecture of the program simulating the system. The efficiency of the method is illustrated by the construction of the two simulation models for the complex systems of various origins: the interactive simulation of the stock exchange and space-time simulation of biological species competition.

A new grid and cloud services simulation for NICA accelerator complex data storage and processing system are described. This system is focused on improving the efficiency of the grid-cloud systems development by using work quality indicators of some real system to design and predict its evolution. For these purpose the simulation program are combined with real monitoring system of the grid-cloud service through a special database. An example of the program usage to simulate a sufficiently general cloud structure, which can be used for more common purposes, is given.

We consider a model of stacked long Josephson junctions (LJJ), which consists of alternating superconducting and dielectric layers. The model takes into account the inductive and capacitive coupling between the neighbor junctions. The model is described by a system of nonlinear partial differential equations with respect to the phase differences and the voltage of LJJ, with appropriate initial and boundary conditions. The numerical solution of this system of equations is based on the use of standard three-point finite-difference formulae for discrete approximations in the space coordinate, and the applying the four-step Runge-Kutta method for solving the Cauchy problem obtained. Designed parallel algorithm is implemented by means of the MPI technology (Message Passing Interface). In the paper, the mathematical formulation of the problem is given, numerical scheme and a method of calculation of the current-voltage characteristics of the LJJ system are described. Two variants of parallel implementation are presented. The influence of inductive and capacitive coupling between junctions on the structure of the current-voltage characteristics is demonstrated. The results of methodical calculations with various parameters of length and number of Josephson junctions in the LJJ stack depending on the number of parallel computing nodes, are presented. The calculations have been performed on multiprocessor clusters HybriLIT and CICC of Multi-Functional Information and Computing Complex (Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna). The numerical results are discussed from the viewpoint of the effectiveness of presented approaches of the LJJ system numerical simulation in parallel. It has been shown that one of parallel algorithms provides the 9 times speedup of calculations.

Small practical value of many numerical methods for solving single-ended systems of linear equations with ill-conditioned matrices due to the fact that these methods in the practice behave quite differently than in the case of precise calculations. Historically, sustainability is not enough attention was given, unlike in numerical algebra ‘medium-sized’, and emphasis is given to solving the problems of maximal order in data capabilities of the computer, including the expense of some loss of accuracy. Therefore, the main objects of study is the most appropriate storage of information contained in the sparse matrix; maintaining the highest degree of rarefaction at all stages of the computational process. Thus, the development of efficient numerical methods for solving unstable systems refers to the actual problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving systems of linear equations, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach consists in minimization of filling the main lines of the multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats. The storage format of sparse matrices has been studied and the advantage of this format consists in possibility of parallel execution any matrix operations without unboxing, which significantly reduces the execution time and memory footprint.

Direct multiplier methods for solving systems of linear equations are best suited for solving problems of large size on a computer — sparse matrix systems allow you to get multipliers, the main row of which is also sparse, and the operation of multiplication of a vector-row of the multiplier according to the complexity proportional to the number of nonzero elements of this multiplier.

As a direct continuation of this work is proposed in the basis for constructing a direct multiplier algorithm of linear programming to put a modification of the direct multiplier algorithm for solving systems of linear equations based on integration of technique of linear programming for methods to select the host item. Direct multiplicative methods of linear programming are best suited for the construction of a direct multiplicative algorithm set the direction of descent Newton methods in unconstrained optimization by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

The work submits new release of the FlowVision software designed for automation of engineering calculations in computational fluid dynamics: FlowVision 3.09.05. The FlowVision software is used for solving different industrial problems. Its popularity is based on the capability to solve complex non-tradition problems involving different physical processes. The paradigm of complete automation of labor-intensive and time-taking processes like grid generation makes FlowVision attractive for many engineers. FlowVision is completely developer-independent software. It includes an advanced graphical interface, the system for specifying a computational project as well as the system for flow visualization on planes, on curvilinear surfaces and in volume by means of different methods: plots, color contours, iso-lines, iso-surfaces, vector fields. Besides that, FlowVision provides tools for calculation of integral characteristics on surfaces and in volumetric regions.

The software is based on the finite-volume approach to approximation of the partial differential equations describing fluid motion and accompanying physical processes. It provides explicit and implicit methods for time integration of these equations. The software includes automated generator of unstructured grid with capability of its local dynamic adaptation. The solver involves two-level parallelism which allows calculations on computers with distributed and shared memory (coexisting in the same hardware). FlowVision incorporates a wide spectrum of physical models: different turbulence models, models for mass transfer accounting for chemical reactions and radioactive decay, several combustion models, a dispersed phase model, an electro-hydrodynamic model, an original VOF model for tracking moving interfaces. It should be noted that turbulence can be simulated within URANS, LES, and ILES approaches. FlowVision simulates fluid motion with velocities corresponding to all possible flow regimes: from incompressible to hypersonic. This is achieved by using an original all-speed velocity-pressure split algorithm for integration of the Navier-Stokes equations.

FlowVision enables solving multi-physic problems with use of different modeling tools. For instance, one can simulate multi-phase flows with use of the VOF method, flows past bodies moving across a stationary grid (within Euler approach), flows in rotary machines with use of the technology of sliding grid. Besides that, the software solves fluid-structure interaction problems using the technology of two-way coupling of FlowVision with finite-element codes. Two examples of solving challenging problems in the FlowVision software are demonstrated in the given article. The first one is splashdown of a spacecraft after deceleration by means of jet engines. This problem is characterized by presence of moving bodies and contact surface between the air and the water in the computational domain. The supersonic jets interact with the air-water interphase. The second problem is simulation of the work of a human heart with artificial and natural valves designed on the basis of tomographic investigations with use of a finite-element model of the heart. This problem is characterized by two-way coupling between the “liquid” computational domain and the finite-element model of the hart muscles.

Multiplicative methods for sparse matrices are best suited to reduce the complexity of operations solving systems of linear equations performed on each iteration of the simplex method. The matrix of constraints in these problems of sparsely populated nonzero elements, which allows to obtain the multipliers, the main columns which are also sparse, and the operation of multiplication of a vector by a multiplier according to the complexity proportional to the number of nonzero elements of this multiplier. In addition, the transition to the adjacent basis multiplier representation quite easily corrected. To improve the efficiency of such methods requires a decrease in occupancy multiplicative representation of the nonzero elements. However, at each iteration of the algorithm to the sequence of multipliers added another. As the complexity of multiplication grows and linearly depends on the length of the sequence. So you want to run from time to time the recalculation of inverse matrix, getting it from the unit. Overall, however, the problem is not solved. In addition, the set of multipliers is a sequence of structures, and the size of this sequence is inconvenient is large and not precisely known. Multiplicative methods do not take into account the factors of the high degree of sparseness of the original matrices and constraints of equality, require the determination of initial basic feasible solution of the problem and, consequently, do not allow to reduce the dimensionality of a linear programming problem and the regular procedure of compression — dimensionality reduction of multipliers and exceptions of the nonzero elements from all the main columns of multipliers obtained in previous iterations. Thus, the development of numerical methods for the solution of linear programming problems, which allows to overcome or substantially reduce the shortcomings of the schemes implementation of the simplex method, refers to the current problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving problems in linear programming, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach is to reduce dimensionality and minimize filling of the main rows of multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats.

As a direct continuation of this work is the basis for constructing a direct multiplicative algorithm set the direction of descent in the Newton methods for unconstrained optimization is proposed to put a modification of the direct multiplier method, linear programming by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

In the article a theory of the implicit iterative line-by-line recurrence method for solving the systems of finite-difference equations which arise as a result of approximation of the two-dimensional elliptic differential equations on a regular grid is stated. On the one hand, the high effectiveness of the method has confirmed in practice. Some complex test problems, as well as several problems of fluid flow and heat transfer of a viscous incompressible liquid, have solved with its use. On the other hand, the theoretical provisions that explain the high convergence rate of the method and its stability are not yet presented in the literature. This fact is the reason for the present investigation. In the paper, the procedure of equivalent and approximate transformations of the initial system of linear algebraic equations (SLAE) is described in detail. The transformations are presented in a matrix-vector form, as well as in the form of the computational formulas of the method. The key points of the transformations are illustrated by schemes of changing of the difference stencils that correspond to the transformed equations. The canonical form of the method is the goal of the transformation procedure. The correctness of the method follows from the canonical form in the case of the solution convergence. The estimation of norms of the matrix operators is carried out on the basis of analysis of structures and element sets of the corresponding matrices. As a result, the convergence of the method is proved for arbitrary initial vectors of the solution of the problem.

The norm of the transition matrix operator is estimated in the special case of weak restrictions on a desired solution. It is shown, that the value of this norm decreases proportionally to the second power (or third degree, it depends on the version of the method) of the grid step of the problem solution area in the case of transition matrix order increases. The necessary condition of the method stability is obtained by means of simple estimates of the vector of an approximate solution. Also, the estimate in order of magnitude of the optimum iterative compensation parameter is given. Theoretical conclusions are illustrated by using the solutions of the test problems. It is shown, that the number of the iterations required to achieve a given accuracy of the solution decreases if a grid size of the solution area increases. It is also demonstrated that if the weak restrictions on solution are violated in the choice of the initial approximation of the solution, then the rate of convergence of the method decreases essentially in full accordance with the deduced theoretical results.

In various applications arise problems modeled by nonlinear partial differential equations on graphs (networks, trees). In order to study such problems and various extreme situations arose in the problems of designing and optimizing networks developed the computational model based on solving the corresponding boundary problems for partial differential equations of hyperbolic type on graphs (networks, trees). As applications, three different problems were chosen solved in the framework of the general approach of network computational models. The first was modeling of traffic flow. In solving this problem, a macroscopic approach was used in which the transport flow is described by a nonlinear system of second-order hyperbolic equations. The results of numerical simulations showed that the model developed as part of the proposed approach well reproduces the real situation various sections of the Moscow transport network on significant time intervals and can also be used to select the most optimal traffic management strategy in the city. The second was modeling of data flows in computer networks. In this problem data flows of various connections in packet data network were simulated as some continuous medium flows. Conceptual and mathematical network models are proposed. The numerical simulation was carried out in comparison with the NS-2 network simulation system. The results showed that in comparison with the NS-2 packet model the developed streaming model demonstrates significant savings in computing resources while ensuring a good level of similarity and allows us to simulate the behavior of complex globally distributed IP networks. The third was simulation of the distribution of gas impurities in ventilation networks. It was developed the computational mathematical model for the propagation of finely dispersed or gas impurities in ventilation networks using the gas dynamics equations by numerical linking of regions of different sizes. The calculations shown that the model with good accuracy allows to determine the distribution of gas-dynamic parameters in the pipeline network and solve the problems of dynamic ventilation management.

Padé approximation is a useful tool for extracting singularity information from a power series. A linear Padé approximant is a rational function and can provide estimates of pole and zero locations in the complex plane. A quadratic Padé approximant has square root singularities and can, therefore, provide additional information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing quadratic Padé approximants as well as some applications. Two algorithms for computing the coefficients in the approximant are discussed: a direct method involving the solution of a linear system (well-known in the mathematics community) and a recursive method (well-known in the physics community). We compare the accuracy of these two methods when implemented in floating-point arithmetic and discuss their pros and cons. In addition, we extend Luke’s perturbation analysis of linear Padé approximation to the quadratic case and identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be identified by the recursive method mentioned above. Another complication with the quadratic approximant arises in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by means of an example. Two concluding applications show the superiority of the quadratic approximant over its linear counterpart: one involving a special function (the Lambert $W$-function) and the other a nonlinear PDE (the continuation of a solution of the inviscid Burgers equation into the complex plane).