


Vol 59, No 3 (2019)
- Year: 2019
- Articles: 13
- URL: https://journal-vniispk.ru/0965-5425/issue/view/11233
Article
Approximation of a Function and Its Derivatives on the Basis of Cubic Spline Interpolation in the Presence of a Boundary Layer
Abstract
The problem of approximate calculation of the derivatives of functions with large gradients in the region of an exponential boundary layer is considered. It is known that the application of classical formulas of numerical differentiation to functions with large gradients in a boundary layer leads to significant errors. It is proposed to interpolate such functions by cubic splines on a Shishkin grid condensed in the boundary layer. The derivatives of a function defined on the grid nodes are found by differentiating the cubic spline. Using this approach, estimates of the relative error in the boundary layer and the absolute error outside of the boundary layer are obtained. These estimates are uniform in a small parameter. The results of computational experiments are discussed.



Classical and Generalized Solutions of a Mixed Problem for a System of First-Order Equations with a Continuous Potential
Abstract
A mixed problem for a first-order differential system with two independent variables and a continuous potential, the corresponding spectral problem for which is the Dirac system, is studied. Using a special transformation of the formal solution and refined asymptotics of the eigenfunctions, the classical solution of the problem is obtained. No excessive conditions on the smoothness of the initial data are imposed. In the case of an arbitrary square summable function, a generalized solution is obtained.



An Approximate Method for Determining the Harmonic Barycentric Coordinates for Arbitrary Polygons
Abstract
A relation for finding the harmonic barycentric coordinates for an arbitrary polygon is obtained. The solution is approximate analytical. In the proposed statement, the harmonic barycentric coordinates are determined in terms of the logarithmic potential of a double layer by solving the Dirichlet problem by the Fredholm method. The approximate nature of the solution is determined by the expansion of the kernel of the integral Fredholm equation of the second kind for the unknown density of potential on the boundary of the domain in the orthogonal Legendre polynomials and the expansion of Green’s function; these expansions are used for the calculation of the potential. An estimate of convergence rate and the error of the solution is obtained. The approximate solutions obtained by the proposed method are compared with the known exact solutions of some benchmark problems.



Mathematical Simulation of Viscous Gas Flows between Two Coaxially Rotating Concentric Cylinders and Spheres
Abstract
A technique for constructing conservative Godunov-type finite-difference schemes for computing viscous gas flows in cylindrical and spherical coordinates is described. Two- and three-dimensional flows in the gap between two coaxially rotating concentric cylinders and spheres are computed. Various types of vortex flows are simulated, which are also typical for an incompressible fluid. The differences from the incompressible case are noted. The results show that cylindrical and spherical Couette flows can be studied within the framework of the mathematical viscous gas model by applying direct numerical simulation with the use of explicit finite-difference schemes.



On a Method for the Probability and Statistical Analysis of the Density of Low Frequency Turbulent Plasma
Abstract
In the framework of the memory flow phenomenology, a model of nonstationary noise is constructed and applied for the stochastic simulation of the time series of the plasma density in a thermonuclear facility. A statistical test that makes it possible to check, at a certain level of significance, the validity of the proposed model based on its agreement with experimental data is presented.



A KP1 Scheme for Acceleration of Inner Iterations for the Transport Equation in 3D Geometry Consistent with Nodal Schemes: Basic Equations and Numerical Results
Abstract
A \(K{{P}_{1}}\) scheme for accelerating the convergence of inner iterations for the transport equation in three-dimensional \(r,\vartheta ,z\) geometry is constructed. This scheme is consistent with the nodal LD (Linear Discontinues) and LB (Linear Best) schemes of the third and fourth orders of accuracy with respect to the spatial variables. To solve the \({{P}_{1}}\) system for acceleration corrections, an algorithm is proposed based on the cyclic splitting method (SM) combined with the tridiagonal matrix algorithm to solve auxiliary systems of two-point equations. A modification of the algorithm for three-dimensional \(x,y,z\) geometry is considered. Numerical examples of using the \(K{{P}_{1}}\) scheme to solve typical radiation transport problems in three-dimensional geometries are given, including problems with a significant role of scattering anisotropy and highly heterogeneous problems with dominant scattering.



Weight Minimization for a Thin Straight Wing with a Divergence Speed Constraint
Abstract
For a thin straight wing satisfying a given constraint on the divergence speed (i.e., the speed above which the twist of the wing leads to its failure), the problem of determining an optimal skin thickness distribution that minimizes the skin mass is considered. The mathematical formulation of the problem is as follows: minimize a linear functional over a set of essentially bounded measurable functions for which the smallest eigenvalue of a Sturm–Liouville problem is no less than a preset value. It is proved that this problem has a unique solution. Since only piecewise smooth thickness distributions satisfy the requirements for applications, the regularity of the optimal solution is analyzed. It turns out that the optimal solution is a Lipschitz continuous function. Additionally, it is shown that the solution depends continuously on a parameter determining the lowest possible divergence speed, i.e., the considered problem is well-posed in the sense of Hadamard. Finally, an iteration method for constructing minimizing sequences converging to an optimal solution in Hölder spaces is proposed and numerical results are presented and discussed.



Conditions for L2-Dissipativity of Linearized Explicit Difference Schemes with Regularization for 1D Barotropic Gas Dynamics Equations
Abstract
Explicit two-level in time and symmetric in space difference schemes constructed by approximating the 1D barotropic quasi-gas-/quasi-hydrodynamic systems of equations are studied. The schemes are linearized about a constant solution with a nonzero velocity, and, for them, necessary and sufficient conditions for the \({{L}^{2}}\)-dissipativity of solutions to the Cauchy problem are derived depending on the Mach number. These conditions differ from one another by at most twice. The results substantially develop the ones known for the linearized Lax–Wendroff scheme. Numerical experiments are performed to analyze the applicability of the found conditions in the nonlinear formulation to several schemes for different Mach numbers.



Comparison of Algorithms for Determining the Thickness of Optical Coatings Online
Abstract
The basic algorithms for determining the thicknesses of layers of deposited multilayer optical coatings are discussed and compared. Using a series of model numerical experiments, the advantage of one of these algorithms—the modified T-algorithm—is demonstrated; this algorithm reduces the influence of the effect of error accumulation in the determined thicknesses of layers.



Mechanism of Moving Particle Aggregates in a Viscous Fluid Subjected to a Varying Uniform External Field
Abstract
A mechanism is proposed for moving aggregates of spherical particles in a viscous fluid, and their dynamics under an applied uniform external field is numerically simulated. A particle aggregate is treated as a set of particles with a charge or dipole moment influenced by both hydrodynamic interaction forces and internal forces retaining the particles in the aggregate. In the absence of an external field, the particles are in the position of minimum interaction energy and the total charge or dipole moment of the system is zero. After applying an external field, the aggregate deforms and, after switching off the field, the aggregate undergoes a restoring process caused by internal forces tending to return it to equilibrium. The relative motion of the aggregate particles gives rise to a viscous flow around the aggregate, which creates a hydrodynamic force shifting the aggregate barycenter in a certain direction with respect to the applied field. The motion of six model aggregates consisting of charged or dipolar particles is numerically simulated. The proposed mechanism of moving the aggregates can be used to control mass transfer in colloidal suspensions.



Solution of Fluid Dynamics Problems in Truncated Computational Domains
Abstract
A mathematical model consisting of quasi-hydrodynamic equations and Dong’s outflow boundary conditions is proposed for solving fluid dynamics problems in a truncated computational domain. A solution algorithm based on finite-element and control-volume methods is developed. The Kovasznay flow and the flow over a backward-facing step in truncated computational domains are numerically simulated. A comparative analysis of the numerical results shows that the proposed mathematical model adequately describes the hydrodynamic flows in a truncated domain.






Frontier Visualization and Estimation of Returns to Scale in Free Disposal Hull Models
Abstract
The free disposal hull (FDH) model of production technology assumes free disposability of all inputs and outputs, but does not make any convexity assumptions. In this paper, we develop an algorithm for the reconstruction of the frontier of the FDH technology that corresponds to its sections by different two-dimensional planes. From a practical perspective, the suggested algorithm is useful for the visualization and exploration of the efficient frontier of the FDH technology. Furthermore, based on the suggested algorithm, we develop a new procedure for the evaluation of returns to scale in the FDH technology. Compared to the existing methods, our approach does not require assessing the efficiency of productive units in the reference technologies, which is known to be a computationally intensive task. Our theoretical results are confirmed by computational experiments in applications with real-life data sets from different industries.


