Home – Home – Jornals – Numerical Analysis and Applications 2023 number 3

The lattice Boltzmann method (LBM) is a numerical scheme for solving fluid dynamics problems. One of the important and actively developing areas of LBM is correct construction of the scheme on non-uniform spatial grids. With non-uniform grids the total number of calculations can be significantly reduced. However, at the moment the construction of an LBM scheme near a boundary of grids with different spatial steps inevitably requires data interpolation, which can reduce the LBM approximation order and lead to violation of conservation laws. In this work, for the first time, we have developed and tested a method for constructing an athermal node-based LBM on non-uniform grids without interpolation, with the same time step for grids of different scales. The method is based on a two-stage transformation of populations corresponding to different on-grid stencils.

We have developed stochastic algorithms to simulate signals detected by a receiver of a laser navigation system designed for safe aircraft landing. Radiant flux and radiance at the receiver, as well as the contribution of radiation of different orders of scattering are estimated by a Monte Carlo method. Computation results show that the proposed algorithms allow one to study the efficiency of the laser navigation system in various conditions.

In this paper, an efficient numerical method which is a collocation method is considered in order to obtain numerical solutions of the KdV-Kawahara equation. The numerical method is based on a finite element formulation and a spline interpolation by trigonometric quintic B-spline basis. Firstly, the KdV-Kawahara equation is split into a coupled equation via an auxiliary variable as υ=u_xxx. Subsequently, a collocation method is applied to the coupled equation together with the forward difference and the Cranck-Nicolson formula. This application leads us to obtain an algebraic equation system in terms of time variables and trigonometric quintic B-spline basis. In order to measure the error between numerical solutions and exact ones, the error norms L₂ and L_∞. are calculated successfully. The results are illustrated by means of two numerical examples with their graphical representations and comparisons with other methods.

A comparative analysis of two algorithms for estimation of weighted mean particle flux - «by particles» and «by collisions» - is made on the basis of test problem solving for a single-speed particle propagation process with scattering and multiplication in a random medium. It is shown that the first algorithm is preferable for a simple estimation of the mean flux and the second one, for estimation of the parameters of a possible superexponential flux growth. Two models of the random medium are considered: a chaotic «Voronoi mosaic» and «a spherically layered mosaic». For a fixed mean correlation radius, the superexponential growth has been stronger for the layered mosaic.

The article considers the problem of calculating the phase probability density function of a phase-shift keying signal received under conditions of distortion and additive noise. This problem is reduced to an inverse problem, namely, to solving an integral equation of the convolution type. The functions included in the integral equation are analyzed. The case of equiprobable symbols, which is important from a practical point of view, is considered separately. Numerical simulation results are presented.

Based on the use of the geometric series theorem, we transform a linear Fredholm integral equation of the second kind defined on a large interval into an equivalent linear system of Fredholm integral equations of the second kind; then, we inflict a refinement in the way the investigated generalised iterative scheme approximates the sought-after solution. By avoiding to inverse a bounded linear operator, and computing a truncated geometric sum of the former's associated sequence of bounded linear operators instead, we notice that our approach furnishes a better performance in terms of computational time and error efficiency.

This article studies a priori error analysis for linear parabolic interface problems with measure data in time in a bounded convex polygonal domain in R2. Both the spatially discrete and the fully discrete approximations are analyzed. We have used the standard continuous fitted finite element discretization for the space while, the backward Euler approximation is used for the time discretization. Due to the low regularity of the data of the problem, the solution possesses very low regularity in the entire domain. A priori error bounds in the L²(L²(Ω))-norm for both the spatially discrete and the fully discrete finite element approximations are derived under minimal regularity with the help of the L²-projection operator and the duality argument. Numerical experiments are performed to underline the theoretical findings. The interfaces are assumed to be smooth for our purpose.

The article solves the problem of determining the temperature on the inner wall of a hollow cylinder. Using a time Fourier transform, the problem is reduced to an ordinary differential equation, with the help of which the Fourier transform of an exact solution of the inverse boundary value problem is found. A projection regularization method is considered, which makes it possible to obtain a stable solution to the problem and an accurate in order of magnitude estimate of the error of the approximate solution. Since high accuracy requirements are imposed on solutions of such problems in numerical calculations, an algorithm is developed to improve the accuracy and reliability of processing the results of thermal test data. To check the performance of the algorithm, test calculations are carried out.