Numerical Analysis and Applications
Numerical Analysis and Applications is an international peer-reviewed journal. The aim of Numerical Analysis and Applications is to show the latest and most important studies in various scientific and engineering fields. The journal deals with the following topics: theory and practice of computational methods, mathematical physics, and other applied fields; mathematical models of elasticity theory, hydrodynamics, gas dynamics, and geophysics; parallelizing of algorithms; models and methods of bioinformatics. The journal welcomes manuscripts from all countries.
Current Issue
Vol 12, No 4 (2019)
- Year: 2019
- Articles: 8
- URL: https://journals.rcsi.science/1995-4239/issue/view/12411
Article
Numerical Solution of a Three-Dimensional Coefficient Inverse Problem for the Wave Equation with Integral Data in a Cylindrical Domain
Abstract
A three-dimensional coefficient inverse problem for the wave equation (with losses) in a cylindrical domain is considered. The data given for its solution are special time integrals of a wave field measured in a cylindrical layer. We present and substantiate an efficient algorithm for solving this three-dimensional problem based on the fast Fourier transform. The algorithm makes it possible to obtain a solution on 512× 512×512 grids in about 1.4 hours on a typical PC without paralleling the calculations. The results of numerical experiments of model inverse problem solving are presented.
Numerical Method for Predicting Hemodynamic Effects in Vascular Prostheses
Abstract
A three-dimensional unsteady periodic flow of blood in xenogenic vascular bioprostheses is simulated by computational fluid dynamics methods. The geometry of the computational domain is based on microtomographic scanning of bioprostheses. To set a variable pressure gradient causing an unsteady flow in the prostheses, personal-specific data of the Doppler echography of the blood flow of a particular patient are used. A comparative analysis of the velocity fields in the flow areas corresponding to three real samples of bioprostheses with multiple stenoses is carried out. In the zones of stenosis and outside of them, the distribution of the near-wall shear stress, which affects the risk factors for thrombosis in the prostheses, is analyzed. An algorithm for predicting the hemodynamic effects arising in vascular bioprostheses is proposed; the algorithm is based on the numerical modeling of the blood flow in these prostheses.
The Walrasian Equilibrium and Centralized Distributed Optimization in Terms of Modern Convex Optimization Methods by an Example of the Resource Allocation Problem
Abstract
The resource allocation problem and its numerical solution are considered. The following is demonstrated: (1) Walrasian price-adjustment mechanism for determining the equilibrium state; (2) decentralized role of prices; (3) Slater’s method for price restrictions (dual Lagrange multipliers); (4) new mechanism for determining equilibrium prices, in which prices are fully controlled by economic agents—nodes (enterprises)—rather than by the Center (Government). In the economic literature, only the convergence of the methods considered is proved. In contrast, this paper provides an accurate analysis of the convergence rate of the described procedures for determining the equilibrium. The analysis is based on the primal-dual nature of the algorithms proposed. More precisely, in this paper, we propose the economic interpretation of the following numerical primal-dual methods of convex optimization: dichotomy and subgradient projection method.
Solving the Pure Neumann Problem by a Finite Element Method
Abstract
This paper deals with the solution of the pure Neumann problem for the diffusion equation by a finite element method. First, an extended generalized formulation of the Neumann problem in the Sobolev space H1(Ω) is derived and investigated. Then a discrete analog of this problem is formulated by using standard finite element approximations of the space H1(Ω). An iterative method for solving the corresponding SLAE is proposed. Some examples of solving model problems are used to discuss the numerical properties of the algorithm proposed.
Improving the Stability of Triangular Decomposition of Ill-Conditioned Matrices
Abstract
An approach to improving the stability of triangular decomposition of a dense positive definite matrix with a large condition number by using the Gauss and Cholesky methods is considered. It is proposed to introduce additions to standard computational schemes with an incomplete inner product of two vectors which is formed by truncating the lower digits of the sum of the products of two numbers. The truncation in the process of decomposition increases the diagonal elements of the triangular matrices by a random number and prevents the appearance of very small numbers during the Gauss decomposition and a negative radical expression in the Cholesky method. The number of additional operations required for obtaining an exact solution is estimated. The results of computational experiments are presented.
Conservation Laws and Other Formulas for Families of Rays and Wavefronts and for the Eikonal Equation
Abstract
In the previous studies, the author has obtained conservation laws for the 2D eikonal equation in an inhomogeneous isotropic medium. These laws are divergent identities of the form div F = 0. The vector field F is expressed through a solution to the eikonal equation (the time field), the refractive index (the equation parameter), and their partial derivatives. Besides that, equivalent conservation laws (divergent identities) were found for families of rays and families of wavefronts in terms of their geometric characteristics. Thus, the geometric essence (interpretation) of the conservation laws obtained for the 2D eikonal equation was found. In this paper, 3D analogs of the results obtained are presented: differential conservation laws for the 3D eikonal equation and conservation laws (divergent identities of the form div F = 0) for families of rays and families of wavefronts, the vector field F expressed through classical geometric characteristics of the ray curves: their Frenet basis (the unit tangent vector, principal normal, and binormal), the first curvature, and the second curvature, or through the classical geometric characteristics of the wavefront surfaces: their normal, principal directions, principal curvatures, Gaussian curvature, and mean curvature. All the results have been obtained on the basis of the general vector and geometric formulas (differential conservation laws and some formulas) obtained for families of arbitrary smooth curves, families of arbitrary smooth surfaces, and arbitrary smooth vector fields.
Simulation of Nonlinear Oscillations in a Micro-Generator of Clock Frequency
Abstract
In this paper, we consider a mathematical model of a new-type micro- generator. The model is based on excitation by electrostatic forces of oscillations of a mobile electrode in a micro gap. The principle of operation of the micro-generator is similar to the well-known theory of clock with a striking escapement mechanism. The difference is that in the equation of motion, the form of the right-hand side takes into account the electrostatic nature of the pulsed action. The numerical analysis shows that with time the bounded oscillations in the phase plane tend to a stable limit cycle, and thus the emerging oscillations are stable towards external perturbations. Studying periodic oscillations in dependence on model parameters relies on the boundary value problem solution for an equation with a discontinuous right-hand side transformed so that to enable application of the method of solution continuation with respect to a parameter. In this way, the domain in the plane of model parameters in which stable limit cycles exist is defined.