System Of Finite Difference Equations Research Articles

Simulation model of an intelligent transportation system for the “smart city” with adaptive control of traffic lights based on fuzzy clustering

This article presents a new simulation model of an intelligent transportation system (ITS) for the “smart city” with adaptive traffic light control. The proposed transportation model, implemented in the AnyLogic, allows us to study the behavior of interacting agents: vehicles (V) and pedestrians (P) within the framework of a multi-agent ITS of the “Manhattan Lattice” type. The spatial dynamics of agents in such an ITS is described using the systems of finite-difference equations with the variable structure, considering the controlling impact of the “smart traffic lights.” Various methods of traffic light control aimed at maximizing the total traffic of the ITS output flow have been studied, in particular, by forming the required duration phases with the use of a genetic optimization algorithm, with a local (“weakly adaptive”) switching control and based on the proposed fuzzy clustering algorithm. The possibilities of optimizing the characteristics of systems for individual control of the behavior of traffic lights under various scenarios, in particular, for the ITS with spatially homogeneous and periodic characteristics, are investigated. To determine the best values of individual parameters of traffic light control systems, such as the phases’ durations, the radius of observation of traffic and pedestrian flows, threshold coefficients, the number of clusters, etc., the previously proposed parallel real-coded genetic optimization algorithm (RCGA type) is used. The proposed method of adaptive control of traffic lights based on fuzzy clustering demonstrates greater efficiency in comparison with the known methods of collective impact and local (“weakly adaptive”) control. The results of the work can be considered a component of the decision-making system in the management of urban services.

Mathematical models of rock mass affected by high pressure gradients

The authors propose to solve the problems connected with high pressure gradients using a model of a linearly elastic body with a structural parameter. The closed finite difference system of equations is formulated for plain strain deformation conditions. The problem on deformation of rock mass in the vicinity of an underground opening affected by mixed-type gravity-and-tectonic stress field is solved. It is demonstrated that addition of the structure in the solution changes the value of the stress concentration coefficient.

Exact method to solve of linear heat transfer problems

When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equations of parabolic and elliptic types, and an exact solution of the system of finite difference equations for a fixed time is obtained. To solve problems with a boundary condition of the first kind on the first coordinate and arbitrary combinations of the first, second and third kinds of boundary conditions on the second coordinate, it is proposed to use the method of straight lines on the first coordinate and ordinary sweep method on the second coordinate. Approximating the equations on the first coordinate, a matrix equation is built relative to the grid functions. Using eigenvalues and vectors of the three-diagonal transition matrix, linear combinations of grid functions are compiled, where the coefficients are the elements of the eigenvectors of the three-diagonal transition matrix. Boundary conditions, and for a parabolic equation, initial conditions are formed for a given combination of grid functions. The resulting one-dimensional differential-difference equations are solved by the ordinary sweep method. From the resulting solution, proceed to the initial grid functions. The method provides a second order of approximation accuracy on coordinates. And the approximation accuracy in time when solving the parabolic equation can be increased to the second order using the central difference in time. The method is used to solve heat transfer problems when the boundary conditions are expressed by smooth and discontinuous functions of a stationary and non-stationary nature, and the right-hand side of the equation represents a moving source or outflow of heat.

Hybrid Fuzzy Differential-Production Model of the Dynamic Drying of a Pellet under Uncertainty

An original hybrid fuzzy differential-production model of the dynamic drying of a multilayer pellet as a dispersed spherical body was developed, which represents the velocity of the localized evaporation front within the pellet being dried. The model is characterized by using a system of partial differential equations of the thermal conductivity and the drying rate with fuzzy parameters representing the interval values of the thermophysical properties of the dispersed pellet particles, by computational operations for reducing the accumulation of the errors of the applied finite-difference methods, and also by replacing the approximating multidimensional finite-difference system of equations by a set of special fuzzy production rules describing the complex dependence of the velocity of the localized evaporation front on the fuzzy thermophysical and heat-and-mass transfer parameters of the drying process. This makes it possible to perform the rapid calculation of the parameters of the process conditions of the resource-and-energy-saving drying of the multilayer pellet with preventing a decrease in the strength of the pellets and their breakability because of the thermal shock. The development of the proposed hybrid fuzzy differential-production model is based on the modern artificial intelligence methods with taking into account the physicochemical features of drying processes.

Finite-difference equations of quasistatic motion of the shallow concrete shells in nonlinear setting

AbstractThe study solves a system of finite difference equations for flexible shallow concrete shells while taking into account the nonlinear deformations. All stiffness properties of the shell are taken as variables,i.e., stiffness surface and through-thickness stiffness. Differential equations under consideration were evaluated in the form of algebraic equations with the finite element method. For a reinforced shell, a system of 98 equations on a 8×8 grid was established, which was next solved with the approximation method from the nonlinear plasticity theory. A test case involved computing a 1×1 shallow shell taking into account the nonlinear properties of concrete. With nonlinear equations for the concrete creep taken as constitutive, equations for the quasi-static shell motion under constant load were derived. The resultant equations were written in a differential form and the problem of solving these differential equations was then reduced to the solving of the Cauchy problem. The numerical solution to this problem allows describing the stress-strain state of the shell at each point of the shell grid within a specified time interval.

Modeling Systems Based on the Assessment of Weak Symmetry Breaking in Reconstructed Attractors

The paper proposes to use the research method for chaotic processes based on the identification of weak symmetry breaking of the restored attractor. It is shown how the calculations results can be used to identify systems. An algorithm for creating finite-difference models has been developed, including: calculating by numerical methods the necessary conditions for the existence of chaos, reconstructing an attractor in a time series, searching for symmetric attractor fragments under conditions of weak symmetry breaking, determining the form of nonlinearities, and parametric identification. The result of the algorithm is a system of finite-difference equations in the state space Criteria for assessing symmetry breaking based on estimates of the divergence of fragments of phase trajectories are introduced. The results of modeling systems with chaotic dynamics are presented.

МЕТОД ПОВЕРХНОСТНЫХ ГАРМОНИК ДЛЯ НЕЙТРОННО-ФИЗИЧЕСКОГО РАСЧЕТА ТРЕХМЕРНОГО ГЕТЕРОГЕННОГО РЕАКТОРА С НЕСИММЕТРИЧНЫМИ ЯЧЕЙКАМИ

The paper addresses the application of the surface harmonics method to building a system of finite difference equations for describing the neutron field of a 3D heterogeneous reactor. The reactor is comprised of cells shaped as right-angled parallelepipeds with non-symmetric contents. Neutrons distribution in the cells is presented as a linear combination of trial functions that satisfy the neutron transport equation and some non-uniform boundary conditions allowing to consider the influence of the cell environment on the neutron distribution within the cell. Even given non-symmetric cells, we can build a system of equations (by sewing the neutron distribution in adjacent cells on joint faces), which, if diffusion approximation is used on the cell faces, are similar in appearance to the finite difference approximation of a group-diffusion equation (using a different method to obtain coefficients of equations, with complete matrices of group neutron diffusion coefficients). Finite-difference equations can also be defined by considering the additional neutron distribution components in cells (trial functions), which are necessary for valid transition to particular cases: two-dimensional and one-dimensional geometry. Due to cell asymmetry, additional components are introduced in equations and formulas for equation coefficients are corrected (correction is similar to the introduction of "discontinuity coefficients" used in foreign codes). As it is shown, the system of equations allows us to avoid using the diffusion approximation in reactor calculations since we can now sew together not only the neutron current density and neutron flux but also higher angular momenta at boundaries of cells without changing the form of equations (increasing the dimension of coefficient matrices and dimension of required vectors). Numerical results are presented on an example of 1D single-velocity test problems.

On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing

In this article, the spectral instability and the associated bifurcations of the shear flows of the 2D quasi-geostrophic equation with a generalized Kolmogorov forcing are investigated. To determine the linear instability of the basic shear flow, we write the corresponding eigenvalue problem as a system of finite difference equations whose nontrivial solutions are expressed in the form of continued fractions. By a rigorous analysis of these continued fractions, we prove the existence of a number Rc such that if the control parameter R, which is proportional to the Reynolds number and the intensity of the curl of forcing, is over Rc, then the basic shear flow loses its stability. To shed light on the bifurcation involved in the loss of stability of the basic shear flow, a natural method is used to reduce the quasi-geostrophic equation to a system of ordinary differential equations. Based on numerical experiments on the coefficients of this reduced system, we show that both supercritical and subcritical Hopf bifurcations occur depending on the frequency of the generalized Kolmogorov forcing. Moreover, we investigate the double Hopf bifurcations which occur at critical aspect ratios. Our results show that in the double Hopf bifurcation case, two periodic solutions, one stable and the other unstable, bifurcate on R>Rc.

Monotone iterative technique for weakly coupled system of finite difference equations of time degenerate parabolic problems

Monotone iterative technique for weakly coupled system of finite difference equations of time degenerate parabolic problems

Application of the analytical discrete velocities method for calculating mass and heat fluxes of rarefied gas in the problem of the Poiseuille flow

Analytical version of the discrete ordinates method is used to for calculating the gas macroparameters in a plane channel in the problem of Poiseuille flow. The system of finite-difference equations that determine the general solution of a homogeneous integro-differential equation after separation of variables is written in a matrix form. Householder algorithm is used for reducing the coefficient matrix to a tridiagonal one. QL algorithm with implicit shifts is applied for searching eigenvalues. The Gauss method with a permutation of the rows is used to determine the coefficients in the expansion of the solution of the required boundary value problem with allowance for the boundary conditions on the channel walls. The software implementation of the algorithm is developed using the C ++ programming language. The mass velocity profile and the heat flow profile of the gas, the flow rate and heat flow through the cross section of the channel are calculated for a wide range of the Knudsen number. The comparison with similar results presented in the open press is performed.

Software Implementation of the discrete ordinates method in the problem of the Poiseuille flow in a plane channel with infinite parallel walls

The algorithm for calculating the gas macroparameters in a channel using the discrete ordinates method was considered. The system of finite-difference equations that determine the general solution of a homogeneous integro-differential equation after separation of variables was written in a matrix form. The coefficient matrix in the resulting system of equations was reduced to a symmetric one and then to a tridiagonal matrix using the Householder algorithm. The search for the eigenvalues of the tridiagonal matrix was performed using a QL algorithm with implicit shifts. The determination of the coefficients in the expansion of the solution of the required boundary value problem with allowance for the boundary conditions on the channel walls was reduced to a system of linear algebraic equations. The Gauss method with a permutation of the rows was used to solve it. The software implementation of the algorithm was carried out using the C++ programming language. The mass velocity profile of the gas and the flow rate through the cross section of the channel in the Poiseuille flow problem were calculated as an application. The comparison with similar results presented in the open press was performed.

Correcting the Semi-Implicit Numerical Scheme Incorporated in the KORSAR Code Two-Liquid Model

Two algorithms used to correct the semi-implicit numerical scheme for time integration of the conservation equations incorporated into the KORSAR/GP computer code two-liquid model are presented. The KORSAR/GP code has been developed jointly by specialists of the Federal State Unitary Enterprise Aleksandrov Research Institute of Technology and Gidropress Experimental Design Office; in 2009, the code was certified at the Federal Service for Environmental, Technological, and Nuclear Supervision (Rostekhnadzor) as applied to numerical safety assessment of VVER-based reactor plants. In the semi-implicit scheme, the convective terms appearing in the phase momentum conservation equations are written in an explicit form. The mass and energy flows of the phases are represented implicitly with respect to phase velocities, and the transferred donor quantities are calculated based on the parameters taken from the previous time layer. Owing to linearization of unsteady and source terms, the linear system of finite difference equations is solved in a noniteration manner. The first of the presented algorithms compensates for numerical imbalances of phase masses and energies resulting from linearization of unsteady terms appearing in the discrete equations and ensures conservativeness of the scheme. The second algorithm introduces correction for the nonphysical redistribution of coolant mass and energy over the computational cells if a change occurs in the phase motion direction within a time step when the scheme of approximating the convective terms becomes “antidonor” with respect to flow. The numerical imbalances and refinements taking into account the correction of donor quantities are computed at each time step according to the proposed correlations and are used for making compensation at the next time step as supplementary sources in the conservation equations. Results from testing the correction algorithms in the KORSAR/GP code are presented. The effectiveness of the imbalance compensation algorithm is confirmed by solving a problem involving the natural circulation loop heating process. The adequacy of the “antidonor” scheme correction algorithm is demonstrated on problems involving steplike initial distribution of the scalar parameters of a stagnant single-phase gaseous or water coolant in a horizontal tube when the flowrate at the tube inlet has different signs at different time steps.

Factor model for assessing the state of the digital economy

The research aims to examine the conceptual model of factors influencing the state of the digital economy and contributing to its effective management. Methodologically the research relies on the cognitive modelling, which develops as an interdisciplinary programme combining the theory of cognition, and artificial intelligence. Cognitive modelling encompasses the stages of constructing a problem field and a cognitive map in the form of digraph; agreeing expert assessments of mutual influence of factors using mathematical statistics; simulation experiment. The problem is solved by the methods that allow for cognitive aspects of percep tion, thinking, explanation, experience, and intuition of researchers. The process of solving the problem is oriented towards involvement of intellectual resources of a specialist to record notions about a problem situation in the form of a formal model, construct a hypothesis on the behaviour of the investigated system, and a forecast of its development. The authors suggest their cognitive model of factors affecting the state of the digital economy and present it in the form of a cognitive map or a weighted directed graph. The model features a particular set of factors with the maximum degree of generalisation and abstraction from concrete subsystems of economy. The authors created a software tool, the algorithm of which was based on the system of finitedifference equations and was implemented using cross-platform technologies that enabled its running without foreign software. This software tool is used to perform a simulation experiment, which demonstrates that the target factor of a cognitive model substantially depends on the control factor “investments”. The researchers determine present conditions for obtaining variants of unstable structure of a cognitive model. Experiments on different combinations of the input data may lead to the desired result of economic situation development

Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations

We consider discrete potentials as controls in systems of finite difference equations which are discretizations of a 1-D Schrodinger equation. We give examples of palindromic potentials which have corresponding steerable initial-terminal pairs which are not mirror-symmetric. For a set of palindromic potentials, we show that the corresponding steerable pairs that satisfy a localization property are mirror-symmetric. We express the initial and terminal states in these pairs explicitly as scalar multiples of vector-valued functions of a parameter in the control.

Accelerated micropolar fluid–flow past an uniformly rotating circular cylinder

In this paper, we formulated the non-steady flow due to the uniformly accelerated and rotating circular cylinder from rest in a stationary, viscous, incompressible and micropolar fluid. This flow problem is examined numerically by adopting a special scheme comprising the Adams-Bashforth Temporal Fourier Series method and the Runge-Kutta Temporal Special Finite-Difference method. This numerical scheme transforms the governing equation into a system of finite-difference equations. This system was further solved numerically by point successive-over-relaxation method. These results were also further extrapolated by the Richardson extrapolation method. This scheme is valid for all values of the flow and fluid-parameters and for all time. Moreover the boundary conditions of the vorticity and the spin at points far from the cylinder are being imposed and encountered too. The results are compared with existing results (for non-rotating circular cylinder in Newtonian fluids). The comparison is good. The enhancement of lift and reduction in drag is observed if the micropolarity effects are intensified. Same is happened if the rotation of a cylinder increases. Furthermore, the vortex-pair in the wake is delayed to successively higher times as rotation parameter increases. In addition, the rotation helps not only in dissolving vortices adjacent to the cylinder and adverse pressure region but also in dissolving the boundary layer separation. Furthermore, the rotation reduces the micropolar spin boundary layer.

On Nonlocal Computation of Eigenfrequencies of Beams Using Finite Difference and Finite Element Methods

In this paper, we show that two numerical methods, specifically the finite difference method and the finite element method applied to continuous beam dynamics problems, can be asymptotically investigated by some kind of enriched continuum approach (gradient elasticity or nonlocal elasticity). The analysis is restricted to the vibrations of elastic beams, and more specifically the computation of the natural frequencies for each numerical method. The analogy between the finite numerical approaches and the equivalent enriched continuum is demonstrated, using a continualization procedure, which converts the discrete numerical problem into a continuous one. It is shown that the finite element problem can be transformed into a system of finite difference equations. The convergence rate of the finite numerical approaches is quantified by an equivalent Rayleigh's quotient. We also present some exact analytical solutions for a first-order finite difference method, a higher-order finite difference method or a cubic Hermitian finite element, valid for arbitrary number of nodes or segments. The comparison between the exact numerical solution and the approximated nonlocal approaches shows the efficiency of the continualization methodology. These analogies between enriched continuum and finite numerical schemes provide a new attractive framework for potential applications of enriched continua in computational mechanics.

A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule

The Poisson–Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for calculating electrostatics of ionic solvated biomolecule. In this paper, a new finite element and finite difference hybrid method is presented to solve PBE efficiently based on a special seven-overlapped box partition with one central box containing the solute region and surrounded by six neighboring boxes. In particular, an efficient finite element solver is applied to the central box while a fast preconditioned conjugate gradient method using a multigrid V-cycle preconditioning is constructed for solving a system of finite difference equations defined on a uniform mesh of each neighboring box. Moreover, the PBE domain, the box partition, and an interface fitted tetrahedral mesh of the central box can be generated adaptively for a given PQR file of a biomolecule. This new hybrid PBE solver is programmed in C, Fortran, and Python as a software tool for predicting electrostatics of a biomolecule in a symmetric 1:1 ionic solvent. Numerical results on two test models with analytical solutions and 12 proteins validate this new software tool, and demonstrate its high performance in terms of CPU time and memory usage.

SMONOTONE METHOD FOR COUPLED SYSTEM OF FINITE DIFFERENCE EQUATIONS OF TIME DEGENERATE PARABOLIC PROBLEMS

The aim of this paper is to develop monotone iterative scheme using the notion of upper-lower solutions for weakly coupled system of finite difference equations which corresponds to weakly coupled system of semilinear time degenerate parabolic Dirichlet initial boundary value problem (IBVP). Using upper and lower solutions as distinct initial iterations, two monotone convergent sequences from linear system are constructed. It is shown that these two sequences converge monotonically from above and below to maximal and minimal solutions respectively which lead to the existence-comparison and uniqueness results for the solution of the discrete nonlinear Dirichlet IBVP. Positivity lemma is the main ingredient used in the proof of these results. AMS Subject Classification: 65Q10

Periodic solutions of nonlinear finite difference systems with time delays

In this paper a coupled system of nonlinear finite difference equations corresponding to a class of periodic-parabolic systems with time delays and with nonlinear boundary conditions in a bounded domain is investigated. Using the method of upper-lower solutions two monotone sequences for the finite difference system are constructed. Existence of maximal and minimal periodic solutions of coupled system of finite difference equations with nonlinear boundary conditions is also discussed. The proof of existence theorem is based on the method of upper-lower solutions and its associated monotone iterations. It is shown that the sequence of iterations converges monotonically to unique solution of the nonlinear finite difference system with time delays under consideration.

Non-equilibrium crystallization in freezing porous media: Numerical solution

Abstract The boundary value problem which arises during heat and moisture transfer in freezing fine-grained porous media under phase transition conditions is solved. It is assumed that the phase transition process occurs with finite rate of the water crystallization. So, the non-instantaneous kinetics is considered. Since the problem is significantly nonlinear the numerical method for the solution is applied. For the approximation of the system of differential equations the implicit two-level finite-difference scheme with central differences for space coordinate and one-side differences for time is used. The finite-difference system of equations is solved by “double sweep method.” It was shown the stability of “double sweep method” and solvability of the problem. Based on the correlation analysis, the dimensionless form for the diffusion coefficient as a function of moisture is obtained and used for the modeling. It is shown that the results for the characteristic distributions — temperature and total moisture, obtained in numerical solution, are in a good agreement with experimental investigations. The effect of the main criteria for the considered process — Lewis and Stefan numbers on the temperature, moisture, ice content and total moisture distributions is discussed. Especial attention was paid on the formation of the kinetic zone and its transformation in the course of non-equilibrium freezing. It was shown that the kinetic zone has a width of about 20–40% of the overall dimension of the system. Therefore the simulation of the phase transition zone as an infinitely thin front in freezing process, which is an approach incorporated in most theoretical models, is not suitable for the non-equilibrium water crystallization processes in fine-grained soils, and thereby conforms the validity of the kinetic approach.