Home – Home – Jornals – Numerical Analysis and Applications 2024 number 1

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 main focus of this paper is an analysis of the semilocal convergence (S.C.) of a three-step Newton-type scheme (TSNTS) used for finding the solution of nonlinear operators in Banach spaces (B.S.). A novel S.C. analysis of the TSNTS is introduced, which is based on the assumption that a generalized Lipschitz condition (G.L.C.) is satisfied by the first derivative of the operator. The findings contribute to the theoretical understanding of TSNTS in B.S. and have practical implications in various applications, such as integral equations further validating our results.

A three-dimensional (3D) matching operator is proposed for a fourth-order accurate solution of a Dirichlet problem of Laplace's equation in a rectangular parallelepiped. The operator is constructed based on homogeneous, orthogonal-harmonic polynomials in three variables, and employs a cubic grid difference solution of the problem for the approximate solution inbetween the grid nodes. The difference solution on the nodes used by the interpolation operator is calculated by a novel formula, developed on the basis of the discrete Fourier transform. This formula can be applied on the required nodes directly, without requiring the solution of the whole system of difference equations. The fourth-order accuracy of the constructed numerical tools is demonstrated further through a numerical example.

The results of modeling the propagation of seismoacoustic waves based on the numerical solution of a direct dynamic problem for a porous medium are considered. The propagation of seismic waves in a porous medium saturated with fluid in the absence of energy loss is described by a system of differential equations of the first order in the 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 posed, the method of complexing the integral Laguerre transform in time with a finite-difference approximation in spatial coordinates is used. The solution algorithm used makes it possible to efficiently carry out calculations when modeling in a complexly constructed porous medium and to investigate the wave effects that arise 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 parallelepiped grids is carried out. Estimates of the influence of the unbalance on the time of solving the problems depending on the number of processors and the number of grid nodes used are given. The results of numerical experiments confirm the theoretical conclusions.

The paper deals with a numerical solution of the wave equation. The solution algorithm uses optimal parameters which are obtained by using Laguerre transform in time for the wave equation. Additional parameters are introduced into a difference scheme of 2nd-order approximation for the equation. The optimal values of these parameters are obtained by minimizing the error of a difference approximation of the Helmholtz equation. Applying the inverse Laguerre transform in the equation for harmonics, a differential-difference wave equation with the optimal parameters is obtained. This equation is difference in the spatial variables and differential in time. An iterative algorithm for solving the differential-difference wave equation with the optimal parameters is proposed. 2-dimensional and 1-dimensional equations are considered. The results of numerical calculations of the differential-difference equations are presented. It is shown that the difference schemes with the optimal parameters give an increase in the accuracy of solving the equations.

In this paper, we provide a new a posteriori error analysis for a 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 a variational discretization method. We first define elliptic reconstructions of numerical solutions and then discuss a posteriori error estimates for all variables.

A problem of variational data assimilation for a sea thermodynamics model is considered, with the aim to reconstruct sea surface heat fluxes taking into account the covariance matrices of input data errors. The sensitivity of some solution functionals to input data in this problem of variational assimilation is studied, and the results of numerical experiments for a model of dynamics of the Baltic Sea are presented.

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.