Home – Home – Jornals – Numerical Analysis and Applications 2020 number 2

The numerical solution of the direct scattering problem for a system of the Zakharov-Shabat equations is considered. Based on the Marchuk identity, a fourth order method of approximation accuracy is proposed. The numerical simulation of the scattering problem is carried out using an example of two characteristic boundary value problems with known solutions. The calculations have confirmed high accuracy of the algorithm proposed, which is necessary in a number of practical applications for optical and acoustic sensing of media in optics and geophysics applied.

We consider an unstudied optimization problem of summing the elements of the two numerical sequences: Y of length N and U of length q ≤ N . The objective of the optimization problem is to minimize the sum of differences of weighted convolutions of sequences of variable lengths (which are not less than q ). In each of the differences, the first convolution is the unweighted autoconvolution of the sequence U nonlinearly expanded in time (by repetitions of its elements), and the second one is the weighted convolution of an expanded sequence with a subsequence of Y . The number of differences is given. We show that the problem is equivalent to that of approximation of the sequence Y by an element of some exponentially sized set of sequences. Such a set consists of all the sequences of length N which include, as subsequences, a given number M of admissible quasiperiodic (fluctuating) repetitions of the sequence U . Each quasiperiodic repetition is generated by the following admissible transformations of the sequence U : (1) shifting U in time, so that the differences between consecutive shifts do not exceed T_max ≤ N , variable expansion of U in time consisting in repeating each element of U , with variable multiplicities of the repetitions. The optimization objective is minimizing the sum of the squares of element-wise differences. We demonstrate that the optimization problem in combination with the corresponding approximation problem are solvable in polynomial time. Specifically, we show that there exists an algorithm which solves the problems in the time O(T³_max M N ). If T_max is a fixed parameter of the problem, then the algorithm running time is O( M N ). In the examples of numerical modeling, we show the applicability of the algorithm to solving applied problems of noise-robust analyzing electrocardiogram-like and photoplethysmogram-like signals.

In this paper a new numerical method with a low-dissipation of a numerical solution, based on a combination of the Godunov method with the Roe scheme, and a piecewise parabolic method on a local stencil is described. The construction of the numerical method is described in considerable detail, the method is validated on the one-dimensional Riemann problem. The results of the numerical simulation of the collision of two relativistic gas spheres are presented.

The propagation of a wave from the point source in the case when the velocity in the medium v is expressed as v = 1/√ y is considered. Exact solutions of the corresponding eikonal equation are obtained and numerically verified. The ambiguity of the solution to this question in the case when the point source is situated at the origin is shown.

This paper considers numerical methods for approaching and simulate the Stokes-Darcy problem, with a new boundary condition. We study herein a robust stabilized fully mixed discretization technique, this method ensures the 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. The well-posedness of the finite element scheme and its convergence analysis are also derived. Finally, the effciency and accuracy of the numerical methods are illustrated by different numerical tests.

A source identification algorithm for the systems of nonlinear ordinary differential equations of the production-destruction type is applied to the inverse problem for the discretized Smoluchowski equation. An unknown source function is estimated by time series of measurements of the specific size particles concentration. Based on an ensemble of adjoint equations solutions, the sensitivity operator is constructed that links the 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 is used with r -pseudoinverse matrices. The eficiency and properties of the algorithm are numerically studied.

A characteristic analysis of the equations of a single-velocity heat-conducting vapor-gas-drop mixture is carried out, in which the interfraction heat exchange is taken into account and their hyperbolicity is shown. The computational formulas of the Godunov method with a linearized Riemann solver are presented with whose use a number of the mixture flows are calculated.

This paper concerns the solution of the inverse boundary value problem for the equation of thermal conductivity and the estimation of the approximate solution error. 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, in the equation of thermal conductivity, the variable was replaced, which led to the synthesis of problems and allowed obtaining an estimate.