Abstract: ABSTRACT A characteristic analysis of the equations of a single-velocity heat-conducting vapor–gas–drop mixture with interfractional heat transfer is carried out. The equations are shown to be hyperbolic. Calculation formulas for a Godunov-type method with a linearized Riemann solver are presented. These formulas are used to calculate some mixture flows. PubDate: 2020-06-01

Abstract: ABSTRACT A source identification algorithm for systems of nonlinear ordinary differential equations of production-destruction type is applied to an inverse problem for a discretized Smoluchowski equation. An unknown source function is estimated by time series of measurements of particle concentrations of a specific size. Based on an ensemble of adjoint equation solutions, a sensitivity operator is constructed that relates perturbations of the sought-for model parameters with perturbations of the measured values. This reduces the inverse problem to a family of quasilinear operator equations. To solve the equations, an algorithm of the Newton–Kantorovich type with \(r\) -pseudoinverse matrices is used. The efficiency and properties of the algorithm are studied numerically. PubDate: 2020-06-01

Abstract: ABSTRACT We consider an unstudied optimization problem of summing elements of two numerical sequences: \(Y\) of length \(N\) and \(U\) of length \(q\leq N\) . The objective of the problem is minimization of the sum of differences of weighted convolutions of sequences of variable length (not less than \(q\) ). In each difference, the first unweighted convolution is the autoconvolution of the sequence \(U\) expanded to a variable length due to multiple repetitions of its elements, and the second one is the weighted convolution of the expanded sequence with a subsequence from \(Y\) . We analyze a variant of the problem with a given input number of differences. We show that the problem is equivalent to that of approximation of the sequence \(Y\) by an element \(X\) of some exponentially-sized set of sequences. Such a set consists of all the sequences of length \(N\) that include as subsequences a given number \(M\) of admissible quasi-periodic (fluctuating) repetitions of the sequence \(U\) . Each quasi-periodic repetition results from the following admissible transformations of the sequence \(U\) : (1) shift of \(U\) by a variable, which do not exceed \(T_{\max}\leq N\) for neighboring repetitions, (2) variable expanding mapping of \(U\) to a variable-length sequence: variable-multiplicity repetitions of elements of \(U\) . The approximation criterion is minimization of the sum of the squares of element-wise differences. We demonstrate that the optimization problem and the respective approximation problem are solvable in a polynomial time. Specifically, we show that there exists an exact algorithm that solves the problem in the time \(\mathcal{O} (T_{\max}^{3}MN)\) . If \(T_{\max}\) is a fixed parameter of the problem, then the time taken by the algorithm is \(\mathcal{O} (MN)\) . In examples of numerical modeling, we show the applicability of the algorithm to solving model applied problems of noise-robust processing of electrocardiogram (ECG)-like and photoplethysmogram (PPG)-like signals. PubDate: 2020-06-01

Abstract: ABSTRACT A low-dissipation numerical method based on a combination of Godunov’s method and a piecewise parabolic method on a local stencil is presented. The construction of the method is described in detail. The method is tested using a one-dimensional problem of breakdown of a discontinuity. The results of a numerical simulation of collision of two relativistic gas spheres are given. PubDate: 2020-06-01

Abstract: ABSTRACT An inverse boundary-value problem for the heat conduction equation is solved, and the error of the approximate solution is estimated. The Fourier transform with respect to time, which allows one to obtain an error estimate, is not applicable to the problem to be solved. Therefore, the variable in the heat conduction equation is replaced, resulting in the synthesis of problems and obtaining the estimate. PubDate: 2020-06-01

Abstract: ABSTRACT The propagation of a wave from a point source when the velocity \(v\) in the medium is expressed as \(v = \frac {1}{\sqrt y}\) is considered. Exact solutions to the corresponding eikonal equation are obtained and numerically verified. It is shown that the solution of this equation when the point source is at the origin of coordinates is not unique. PubDate: 2020-06-01

Abstract: ABSTRACT This paper considers numerical methods for approximating and simulating the Stokes–Darcy problem, with a new boundary condition. We study a robust stabilized fully mixed discretization technique. This method ensures stability of the finite element scheme and does not use any Lagrange multipliers to introduce a stabilization term in the temporal Stokes–Darcy problem discretization. A correct finite element scheme is obtained and its convergence analysis is done. Finally, the efficiency and accuracy of these numerical methods are illustrated by different numerical tests. PubDate: 2020-06-01

Abstract: ABSTRACT A numerical solution of the direct scattering problem for the system of Zakharov–Shabat equations is considered. Based on the Marchuk identity, a method of fourth-order approximation accuracy is proposed. A numerical simulation of the scattering problem is made using two characteristic boundary value problems with known solutions as an example. The calculations have shown that the algorithm has high accuracy, which is necessary in many practical applications of optical and acoustic sensing in applied optics and geophysics. PubDate: 2020-06-01

Abstract: ABSTRACT This paper deals with a parallel implementation of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm to numerically solve the Navier–Stokes system of equations for viscous incompressible flows. A mechanism of interprocess communication using a mesh decomposition with virtual cells and an algebraic multigrid method is proposed. A method of distributed matrix storage and an algorithm for matrix-vector operations reducing the number of interprocess communications are presented. The results of a series of numerical experiments on structured and unstructured grids (including a problem of external aerodynamics) are presented. Based on the results obtained, an analysis of the influence of multigrid solver settings on the overall efficiency of the algorithm is made. It is shown that the parallel algorithm based on an algebraic multigrid technique proposed for the SIMPLE method makes it possible to efficiently calculate problems on hundreds of processors. PubDate: 2020-02-01

Abstract: ABSTRACT In this article, we discuss a fourth-order accurate scheme based on non-polynomial splines in tension approximations for solving quasi-linear parabolic partial differential equations (PDEs). The proposed numerical method requires only two half-step points and a central point on a uniform mesh in spatial direction. This method is derived directly from the continuity condition for the first-order derivative of the non-polynomial tension spline function. The stability of the scheme is discussed using a model linear PDE. The method is applicable for solving singular parabolic problems in polar systems. The proposed method is tested on the generalized Burgers–Huxley equation, generalized Burgers–Fisher equation, and Burgers’ equations in polar coordinates. PubDate: 2020-02-01

Abstract: ABSTRACT In this paper we discuss a priori error estimates and superconvergence of splitting positive definite mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element functions, and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, state variables, and co-state variables. Second, we obtain a superconvergence result for the control variable. PubDate: 2020-02-01

Abstract: ABSTRACT All possible symmetric two-level difference schemes on arbitrary extended stencils are considered for the Schrödinger equation and for the heat conduction equation. The coefficients of the schemes are found from conditions under which the maximum possible order of approximation with respect to the main variable is attained. A class of absolutely stable schemes is considered in a set of maximally exact schemes. To investigate the stability of the schemes, the von Neumann criterion is verified numerically and analytically. It is proved that the schemes are absolutely stable or unstable depending on the order of approximation with respect to the evolution variable. As a result of the classification, absolutely stable schemes up to the tenth order of accuracy with respect to the main variable have been constructed. PubDate: 2020-02-01

Abstract: ABSTRACT A form of Rosenbrock-type methods optimal in terms of the number of non-zero parameters and computational costs per step is considered. A technique of obtaining \((m, k)\)-methods from some well-known Rosenbrock-type methods is justified. Formulas for transforming the parameters of \((m,k)\)-schemes and for obtaining a stability function are given for two canonical representations of the schemes. An \(L\)-stable \((3, 2)\)-method of order 3 is proposed, which requires two evaluations of the function: one evaluation of the Jacobian matrix and one \(LU\)-decomposition per step. A variable step size integration algorithm based on the \((3,2)\)-method is formulated. It provides a numerical solution for both explicit and implicit systems of ODEs. Numerical results are presented to show the efficiency of the new algorithm. PubDate: 2020-02-01

Abstract: ABSTRACT This paper deals with a difference scheme of second-order approximation using Laquerre transform for the one-dimensional Maxwell equations. Supplementary parameters are introduced into this difference scheme. These parameters are obtained by minimizing the error of a difference approximation for a Helmholtz equation. The optimal parameters do not depend on the step size and the number of nodes in the difference scheme. It is shown that the use of the Laguerre decomposition method allows obtaining higher accuracy of approximation of the equations in comparison with similar difference schemes when using the Fourier decomposition method. The second-order finite difference scheme with the parameters is compared to a fourth-order difference scheme in two cases: The use of the optimal difference scheme when solving a problem of electromagnetic pulse propagation in an inhomogeneous medium yields a solution accuracy comparable to that obtained with the fourth-order difference scheme. When solving an inverse problem the second-order difference scheme allows obtaining higher solution accuracy as compared to the fourth-order difference scheme. In these problems the second-order difference scheme with the supplementary parameters has decreased the calculation time by 20–25% as compared to the fourth-order difference scheme. PubDate: 2020-02-01

Abstract: ABSTRACT This paper deals with convergence of spectral and conditional spectral models to simulate the stochastic structure of sea surface undulations and rogue ocean waves. Convergence of spatial-temporal and spatial models is studied in particular. PubDate: 2020-02-01

Abstract: 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. PubDate: 2019-10-01

Abstract: 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. PubDate: 2019-10-01

Abstract: 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. PubDate: 2019-10-01

Abstract: 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. PubDate: 2019-10-01