For a 3-dimensional dynamical system considered as a model of a gene network with nonlinear degradation of its components, the uniqueness of an equilibrium point is proved. Using approaches of qualitative theory of ordinary differential equations, we find conditions of existence of a cycle of this system and describe an invariant domain which contains all such cycles in the phase portrait. Numerical experiments with trajectories of this system are conducted.
The results of seismoacoustic wave propagation modeling based on a numerical solution of a direct dynamic problem for a porous medium are considered. The propagation of seismic waves in a porous medium saturated with a fluid in the absence of energy loss is described by a system of first-order differential equations in a Cartesian coordinate system. The initial system is written as a hyperbolic system in terms of the velocities of the elastic host medium, the velocity of the saturating fluid, the components of the stress tensor, and the pressure of the fluid. For the numerical solution of the problem, a method of complexing the integral Laguerre transform in time with a finite-difference approximation in the spatial coordinates is used. The solution algorithm makes it possible to efficiently carry out calculations of modeling in a complexly constructed porous medium and investigate wave effects in such media.
A study of the influence of unbalancing the processor load in parallelization of solutions of (3D) boundary value problems on quasi-structured parallelepipedal grids is carried out. Estimates of the influence of the unbalance on the time of solving the problems versus the number of the processors and the number of the grid nodes used are given. The results of numerical experiments confirm the theoretical conclusions.
An inverse problem of reconstructing a coefficient in the differential equation of a model of development for a homogeneous biological population of organisms structured by age is considered. The model takes into account the impact of migration flows on population size changes. Conditions are established to ensure the uniqueness of the solution of the inverse problem. A brief overview of algorithms for the numerical solution of the inverse problem is provided.
In this paper, we provide a new a posteriori error analysis for linear finite element approximation of a parabolic integro-differential optimal control problem. The state and co-state are approximated by piecewise linear functions, while the control variable is discretized by variational discretization method. We first define the elliptic reconstructions of numerical solutions and then discuss a posteriori error estimates for all variables.
A difference scheme of the predictor–corrector type is constructed for solving nonlinear dispersive equations of wave hydrodynamics with an increased order of approximation of the dispersion relation, based on splitting of the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. A dissipation and dispersion analysis of the new scheme is performed, a condition for its stability is obtained, and a formula for the phase error is written and analyzed. Parameters are found at which the phase characteristics of the difference scheme, the nonlinear-dispersive model approximated by it, and the full model of potential flows have the same order of accuracy.
The Lax–Friedrichs scheme is traditionally considered an alternative to the Godunov scheme, since it does not require solving the Riemann problem. In the equations of special relativistic hydrodynamics, the speed of light is a natural limitation of the wave propagation speed. The use of such an upper estimate of the slopes of characteristics in the schemes of Roe, the Rusanov type, or the Harten–Lax–van Leer family leads to a construction equivalent to the Lax–Friedrichs scheme. Due to the absolute robustness of the scheme, a number of software implementations have been developed on its basis for modeling relativistic gas flows. In this paper, we propose a piecewise parabolic reconstruction of the physical variables to reduce dissipation of the numerical method. The use of such a reconstruction in the Lax–Friedrichs scheme allows us to obtain an absolutely robust simple scheme of high-order accuracy on smooth solutions and with small dissipation at the discontinuities. The computational experiments carried out in the article confirm these properties of the scheme.
A new iterative method for modeling of non-Gaussian random vectors with given marginal distributions and a covariance matrix is proposed in this paper. The algorithm is compared with another iterative algorithm for modeling of non-Gaussian vectors, based on reordering of a sample of independent random variables with given marginal distributions. Our numerical studies show that both algorithms are equivalent in terms of the accuracy of reproduction of a given covariance matrix, but the offered algorithm turns out to be more efficient in terms of memory usage and, in many cases, is faster than the other one.
A new correlative-grid approximation for a homogeneous isotropic random field of density is introduced for effective numerical-analytical investigation of overexponential growth of the mean flux of particles with multiplication in a random medium. In this case, the complexity of realization of a particle trajectory is independent of the correlation scale. For the correlative-grid approximation, the possibility of a Gaussian asymptotics of the average rate of particle multiplication is proved for a random field of limited density. It ensures a superexponential growth of the flux in some initial time interval. An estimate of further overexponential flux growth is constructed based on some test computations.
To calculate flows of a gas-liquid mixture, a modified inverse method of characteristics is proposed. An additional fractional time step is introduced in its algorithm, which makes it possible to carry out calculations with a large time step without loss of accuracy and stability. A formulation of boundary conditions on curvilinear walls is discussed in relation to a multidimensional nodal method of characteristics which is based on splitting along the coordinate directions of the original system of equations into a number of one-dimensional subsystems. For the boundary points located on curvilinear impenetrable surfaces, a calculation method based on a method of fictitious nodes is proposed. When testing the modified method, a supersonic interaction of a homogeneous dispersed flow with a barrier is calculated for a flow regime with an attached shock wave. Problems of steady mixture flows near an external obtuse angle, as well as near a cone, which are analogues of Prandtl–Meyer and Busemann flows in gas dynamics, are solved. The calculation results are compared with available self-similar solutions, and a satisfactory agreement is reached.
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