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The paper presents a computational algorithm for solving an optimal control problem formulated in terms of mean-field game theory. The equilibrium of the mean field leads to a coupled system of two parabolic partial differential equations: Fokker-Plank-Kolmogorov and Hamilton-Jacobi-Bellman ones. We extend the conventional model by incorporating a mechanism for external (crisis) influence on the behavior of agents. The article is devoted to the discrete approximation of such a differential statement. The constructed algorithm is implemented to the task of heating insulation of houses. [ABSTRACT FROM AUTHOR]

The paper presents a new mathematical model of the convection-diffusion process with 'memory along the flow path'. It is described by one-dimensional initial-boundary value problem with a fractional derivative along the characteristic curve of convection operator. The constructed model satisfies the local and global conservation laws. The finite-difference approximation of the problem is constructed on the base of Lagrange approach. The stability and discrete conservation laws are proven for algorithmic realization of this approximation. [ABSTRACT FROM AUTHOR]

The paper presents a computational algorithm for solving problems formulated in terms of Mean Field Game theory with restriction on the control resource. The Mean Field equilibrium leads to a coupled system of two parabolic partial differential equations: Fokker-Planck-Kolmogorov and Hamilton-Jacobi-Bellman ones. The article is devoted to the discrete approximation of these equations. In addition, contrary to other studies, the algorithm constructed here is applicable to problem with more general control function. This algorithm is applied to the problem of heating insulation. [ABSTRACT FROM AUTHOR]

The paper presents a computational algorithm for solving the problem of pricing an American put-option described by the Black-Sholes equation. Firstly, two changes of variables are presented. First change is used to eliminate a discontinuity of solution derivatives. The second one transforms the equation on the domain with free boundary to the equation with an unknown coeffcient on a domain with fixed boundary. Secondly, the finite-difference approximation of the obtained equation is fulfilled including boundary condition at infinity. Finally, the computational algorithm is presented for solving discrete analogue of parabolic equation with iteration of the secant method at each time layer to found an unknown coeffcient with the help of the additional boundary condition. [ABSTRACT FROM AUTHOR]

The paper presents a computational algorithm for solving problems formulated in terms of Mean Field Game theory with diffusion expressed by fractional derivative. The Mean Field equilibrium leads to a coupled system of two parabolic partial differential equations: modified Fokker-Planck-Kolmogorov and Hamilton-Jacobi-Bellman equations with the highest derivative in space of fractional order less than 2. The article is devoted to the discrete approximation of these equations and the computational algorithm for solving discrete analogues. [ABSTRACT FROM AUTHOR]

The paper presents a computational scheme for solving economic problems formulated in terms of Mean Field Game theory. The equilibrium of the Mean Field leads to a coupled system of two parabolic partial differential equations: Fokker-Planck-Kolmogorov and Hamilton-Jacobi-Bellman ones. The article is devoted to the discrete approximation of these equations and the application of the Mean Field theory directly at the discrete level. Contrary to other studies, the monotonic algorithm constructed here is applicable to problem with more general control function. This algorithm is applied to the problem of heating insulation. [ABSTRACT FROM AUTHOR]

The paper gives a review of using the semi-Lagrangian approximation depending on the fulfillment of conservation laws for the transfer operator. We begin with approximations of the one-dimensional transfer equation and a parabolic one as simple methodological examples. For two-dimensional problems, first we apply one-dimensional approximations in two directions separately. Then we present another combined approximation along trajectories of the transfer operator. For parabolic and transfer equations, the principles of constructing discrete analogues are demonstrated for three different conservation laws of transfer operator (or the requirements of stability in the related discrete norms similar to the L1, L2, L∞ - norms). It is significant that different conservation laws yield distinct difference problems as well as distinct ways to justify their stability. [ABSTRACT FROM AUTHOR]

The numerical methods are presented for solving economic problems formulated in the Mean Field Game (MFG) form. The mean-field equilibrium (i.e., the Nash equilibrium for an infinite number of players) leads to the coupled system of two parabolic partial differential equations: the Hamilton-Jacobi-Bellman-Isaacs equation and the Fokker-Planck-Kolmogorov one. The description is focused on the discrete approximation of these equations and on the application of the MFG theory directly at discrete level. This approach results in an efficient algorithm for finding the corresponding grid control function. Contrary to difference schemes with directed differences used by other authors, here the semi-Lagrangian approximation is applied which improves some properties of a discrete problem of this type. This implies the fast convergence of an iterative algorithm for the monotone minimization of the cost functional. [ABSTRACT FROM AUTHOR]

In the paper, a mathematical model and a numerical algorithm are proposed for modeling an air flow. The proposed model is based on the time-dependent Navier-Stokes equations for viscous heat-conducting gas. The energy equation and the state equations are modified to account for two kinds of 'internal' energy. The first one is the usual translational and rotational energy of molecules which defines the thermodynamical temperature and the pressure. The second one is the subgrid energy of small turbulent eddies. A numerical algorithm is proposed for solving the formulated initial-boundary value problem as a combination of the semi-Lagrangian approximation for Lagrange transport derivatives and the conforming finite element method for other terms. A numerical example illustrates these approaches. [ABSTRACT FROM AUTHOR]

In the paper, a semi-Lagrangian approximation is presented for the numerical solution of the two-dimensional timedependent Navier-Stokes equations for viscous heat-conducting gas. In each equation, a combination of three first-order derivatives describing the transfer of a corresponding substance (density, velocity components, or internal energy) along trajectories is interpreted as the "transfer derivative" in the transfer direction. The other terms of the equations are written in the Euler form. On the sought-for time level, the standard conforming finite element method is realized for them with the linear elements on triangles and the bilinear ones on rectangles. The stencil adaptation along trajectories enables us to avoid the Courant-Friedrichs-Lewy upper limit which describes the dependence of the time step on the mesh-size of the space triangulation. At the end of the paper, a numerical example illustrates the implementation of the described algorithms. [ABSTRACT FROM AUTHOR]

In the paper, a mathematical model and a numerical algorithm are proposed for modeling the complex of phenomena which accompany the passage of a friable asteroid-comet body through the Earth's atmosphere: the material ablation, the dissociation of molecules, and the radiation. The proposed model is constructed on the basis of the Navier-Stokes equations for viscous heat-conducting gas with an additional equation for the motion and propagation of a friable lumpy-dust material in air. The energy equation is modified for the relation between two its kinds: the usual energy of the translation of molecules (which defines the temperature and pressure) and the combined energy of their rotation, oscillation, electronic excitation, dissociation, and radiation. For the mathematical model of atmosphere, the distribution of density, pressure, and temperature in height is taken as for the standard atmosphere. An asteroid-comet body is taken initially as a round body consisting of a friable lumpy-dust material with corresponding density and significant viscosity which far exceed those for the atmosphere gas. A numerical algorithm is proposed for solving the initial-boundary problem for the extended system of Navier-Stokes equations. The algorithm is the combination of the semi-Lagrangian approximation for Lagrange transport derivatives and the conforming finite element method for other terms. The implementation of these approaches is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

In the paper, a semi-Lagrangian approximation is presented for the numerical solution of the two-dimensional timedependent Navier-Stokes equations for viscous heat-conducting gas. In each equation, a combination of three first-order derivatives describing the transfer of a corresponding substance (density, velocity components, or internal energy) along trajectories is interpreted as the "transfer derivative" in the transfer direction. The other terms of the equations are written in the Euler form. On the sought-for time level, the standard conforming finite element method is realized for them with the linear elements on triangles and the bilinear ones on rectangles. The stencil adaptation along trajectories enables us to avoid the Courant-Friedrichs-Lewy upper limit which describes the dependence of the time step on the mesh-size of the space triangulation. At the end of the paper, a numerical example illustrates the implementation of the described algorithms. [ABSTRACT FROM AUTHOR]

In the paper, a mathematical model and a numerical algorithm are proposed for modeling the complex of phenomena which accompany the passage of a friable asteroid-comet body through the Earth's atmosphere: the material ablation, the dissociation of molecules, and the radiation. The proposed model is constructed on the basis of the Navier-Stokes equations for viscous heat-conducting gas with an additional equation for the motion and propagation of a friable lumpy-dust material in air. The energy equation is modified for the relation between two its kinds: the usual energy of the translation of molecules (which defines the temperature and pressure) and the combined energy of their rotation, oscillation, electronic excitation, dissociation, and radiation. For the mathematical model of atmosphere, the distribution of density, pressure, and temperature in height is taken as for the standard atmosphere. An asteroid-comet body is taken initially as a round body consisting of a friable lumpy-dust material with corresponding density and significant viscosity which far exceed those for the atmosphere gas. A numerical algorithm is proposed for solving the initial-boundary problem for the extended system of Navier-Stokes equations. The algorithm is the combination of the semi-Lagrangian approximation for Lagrange transport derivatives and the conforming finite element method for other terms. The implementation of these approaches is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

Computational aerodynamics is a key technology in aircraft design which is ahead of physical experiment and complements it. Of course all three components of computational modeling are actively developed: mathematical models of real aerodynamic processes, numerical algorithms, and high-performance computing. The most impressive progress has been made in the field of computing, though with a considerable complication of computer architecture. Numerical algorithms are developed more conservative. More precisely, they are offered and theoretically justified for more simple mathematical problems. Nevertheless, computational mathematics now has amassed a whole palette of numerical algorithms that can provide acceptable accuracy and interface between modern mathematical models in aerodynamics and high-performance computers. A significant step in this direction was the European Project ADIGMA whose positive experience will be used in International Project TRISTAM for further movement in the field of computational technologies for aerodynamics. This paper gives a general overview of objectives and approaches intended to use and a description of the recommended four-stage computer technology. [ABSTRACT FROM AUTHOR]

We present the semi-Lagrangian approximation of transfer operator for three-dimensional convection-diffusion problem with corresponding initial and boundary conditions. This problem describes, for instance, a transfer of a substance with diffusion. To construct numerical method, we decompose operator of this equation into two parts. The first one is the transfer operator. The second part is the elliptic diffusion terms. To approximate the first part, we use conservative semi-Lagrangian approximation connecting two integrals of solution at neighboring time levels. To compute integral at the previous time level, we approximate integral domain by 48 tetrahedrons and interpolate solution by trilinear functions. The second part is approximated by conventional finite differences. Finally, we combine approximations of two parts to get a system of linear equations at each time level. Matrix of this system at each time level is the symmetric M-matrix. The proposed approximation has the first-order convergence that is confirmed by computational experiments. [ABSTRACT FROM AUTHOR]

In the paper, the semi-Lagrangian method is considered for the numerical solution of the advection problem. A numerical solution is constructed as a piecewise constant function on a rectangular grid. The proposed method is stable and gives an approximate solution with the first order of accuracy. To reduce the effect of smoothing an approximate solution because of numerical viscosity, a mesh refinement is applied in the vicinity of large gradients of the approximate solution. The localization of the smoothing effect is illustrated by a numerical example. In contrast to the traditional Eulerian schemes, semi-Lagrangian algorithms do not involve a time step restriction. [ABSTRACT FROM AUTHOR]

In the paper, we propose new biquadratic and bicubic Hermite elements on a rectangle. These elements involve the values of a function and its second-order derivatives as degrees of freedom. The proposed elements are more efficient than Lagrange elements of the same degree. Numerical results confirm the theoretical estimates of convergence. [ABSTRACT FROM AUTHOR]