Vol 59, No 6 (2019)

Given an ensemble of numerical solutions generated by different algorithms that are guaranteed to have different errors, the triangle inequality is used to find a neighborhood of a numerical solution that contains the true one. By analyzing the distances between the numerical solutions, the latter can be ranged according to their error magnitudes. Numerical tests for the two-dimensional compressible Euler equations demonstrate the possibility of comparing the errors of different methods and determining a domain containing the true solution.

Various continuous linear functionals of integral curves of a dynamical system are optimally estimated using incorrect additive observations of this system. A numerical-analytical method for analyzing the behavior of the system is developed in the case when observations contain not only counts of an integral curve and fluctuation noise, but also counts of singular disturbance. The method yields optimal unbiased and invariant (with respect to the given noise) estimates without using conventional state space extension. The random and systematic errors are analyzed, and illustrative examples are given.

The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step of choosing the direction of the fastest descent at each iteration of the greedy algorithm. We show that these values give upper bounds for the difference between the values of the objective function in the current state and the optimal point. Since the value of the objective function at the optimal point is not known in advance, the current values of the duality gap can be used, for example, in the stopping criteria for the greedy algorithm. In the paper, we find estimates of the duality gap values depending on the number of iterations for the weak greedy algorithms under consideration.

We study existence and dynamics of bounded traveling wave solutions to generalized \(\theta \)-equation from the perspective of dynamical systems. We obtain bifurcation of traveling wave solutions for the equation, prove the existence of several types of bounded traveling wave solutions, including solitary wave solutions, periodic wave solutions, peakons, periodic cusp waves, compactons and kink-like (antikink-like) waves, and derive some of their exact expressions. Most importantly, we confirm abundant dynamical behaviors of the traveling wave solutions to the equation, which are summarized as follows: (1) We confirm that three types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, the composed homoclinic orbit which is comprised of three heteroclinic orbits of the associated system, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of the associated system. (2) We confirm that four types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, the periodic orbit surrounding two connected homoclinic orbits, the composed periodic orbit which is comprised of two heteroclinic orbits of the associated system, and the homoclinic orbit of the associated system which is tangent to the singular line at the singular point of the associated system. (3) We confirm that two types of orbits correspond to periodic cusp waves, that is, the semiellipse orbit surrounding a center, and the semiellipse-like orbit surrounding two connected homoclinic orbits. (4) We confirm that two families of periodic orbits, which surround two connected homoclinic orbits and are comprised of two heteroclinic orbits of associated system, respectively, and the composed homoclinic orbit, which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system, have envelope.

A linear hypersingular integral equation is considered on a surface (closed or nonclosed with a boundary). This equation arises when the Neumann boundary value problem for the Laplace equation is solved by applying the method of boundary integral equations and the solution is represented in the form of a double-layer potential. For such an equation, a numerical scheme is constructed by triangulating the surface, approximating the solution by a piecewise linear function, and applying the collocation method at the vertices of the triangles approximating the surface. As a result, a system of linear equations is obtained that has coefficients expressed in terms of integrals over partition cells containing products of basis functions and a kernel with a strong singularity. Analytical formulas for finding these coefficients are derived. This requires the computation of the indicated integrals. For each integral, a neighborhood of the singular point is traversed so that the system of linear equations approximates the integrals of the unknown function at the collocation points in the sense of the Hadamard finite part. The method is tested on some examples.

The transmission problem describing a steady-state temperature distribution in a plane consisting of two half-planes occupied by different materials with exponential internal thermal conductivities with a single finite crack along the interface is considered. The compatibility conditions for the boundary functions are formulated under which the problem has a unique classical solution. Closed-form representations of the classical solution are found. The weak solution to the problem is studied without making additional assumptions, and asymptotic expansions are constructed for the weak solution and its first derivatives near the ends of the crack.

An important problem in the numerical simulation of turbulent heat exchange in fluids is accurate prediction of hydrodynamic characteristics of the flow in the boundary layer, which requires a fine grid near rigid surfaces. In applications, it is not always possible to have a fine grid and the use of a coarser grid results in significant loss of accuracy. A well-known approach to improving the accuracy of the numerical simulation of the boundary layer is the use of universal wall functions for computing the friction and thermal flux. In this paper, we consider the known wall functions for computing the thermal flux. The accuracy of these functions in problems of turbulent nonisothermal flow of fluid is studied. These are the flow in a plane channel, Couette flow, and flow along a heated plate. Each of these problems is solved on grids with various near wall resolutions. The results of solving these problems provide a basis for estimating the accuracy of the wall functions used for solving them. It is shown that the wall functions considered in this study yield nonmonotonic convergence of the results as the grid is refined.

An algorithm is constructed for numerical study of the plane viscous flow (in the Stokes approximation) between two arbitrarily moving cylinders of arbitrary cross section. The mathematical model of the problem is described by the biharmonic equation, which is reduced, with the help of the Goursat representation, to a system of equations for two harmonic functions. The system is solved numerically by applying the boundary element method without saturation. As a result, the desired biharmonic stream function and the velocity field are determined. Test examples are used to compare the results with well-known exact solutions for circular cylinders.