Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki

4 февраля 2023 г. исполнилось 90 лет доктору физико-математических наук, профессору Евгению Михайловичу Шахову – одному из лидеров современной механики разреженного газа.

The S-model kinetic equation is used to study the rarefied gas flow from a high-pressure tank to a low-pressure one through a flat membrane with a finite number of pores. The kinetic equation is solved numerically using a second-order accurate implicit conservative method implemented in the in-house code Nesvetay. For transitional and continuum flow regimes, numerical solutions of the compressible Navier–Stokes equations are obtained. The gas flow rate through the system of pores and the forces acting on the membrane bars are investigated as functions of the Knudsen number (Kn) at a pressure ratio of 2 : 1 in the tanks. The features of the flow field near the membrane and away from it are described.

Model equations approximating the system of Boltzmann equations for a multicomponent gas are investigated. Methods for determining parameters in relaxation terms corresponding to cross-collision integrals are analyzed. Numerical solutions based on three model systems and the Boltzmann equations are compared as applied to the following problems: relaxation of a mixture to equilibrium, shock wave structure, and the dynamics of a vapor-gas cloud generated by pulsed laser irradiation of a target. It is shown that the parameters in the relaxation operators influence the degree of difference in the solutions produced by the various models.

A data parallelization algorithm for the direct simulation Monte Carlo method for rarefied gas flows is considered. The scaling of performance of the main algorithm procedures are analyzed. Satisfactory performance scaling of the parallel particle indexing procedure is shown, and an algorithm for speeding up the operation of this procedure is proposed. Using examples of solving problems of free flow and flow around a cone for a 28-core node with shared memory, an acceptable speedup of the entire algorithm was obtained. The efficiency of the data parallelization algorithm and the computational domain decomposition algorithm for free flow is compared. Using the developed parallel code, a study of the supersonic rarefied flow around a cone is carried out.

This paper overviews the state of the art in the study of nonequilibrium gas flows with nonclassical transport, in which the Stokes and Fourier laws are violated (and, accordingly, the Chapman–Enskog method is inapplicable). For a reliable validation of anomalous transport effects, we use computational methods of different nature: the direct solution of the Boltzmann equation and direct simulation Monte Carlo. Nonclassical anomalous transport is manifested on scales of 5–10 mean free paths, which confirms the fact that a highly nonequilibrium flow is a prerequisite for the detection of the effects. Two-dimensional flow problems are considered, namely, the supersonic flow over a flat plate in the transient regime and the supersonic flow through membranes (lattices), where the flow behind the lattice corresponds to the spatially nonuniform relaxation problem. In this region, nonequilibrium distributions demonstrating anomalous transport are formed. The relationship of the effect with the second law of thermodynamics is discussed, the possibilities of experimental verification are considered, and the prospects of creating new microdevices on this basis are outlined.

Flow around a body moving at a high subsonic velocity in a tube filled with rarefied gas is studied. This aerodynamic problem is considered as applied to the task of designing a high-speed vacuum transport at finite Knudsen numbers. Parameters that are close to target characteristics of such systems are chosen, more precisely, speed of about 1000 km/h, significant transverse size of the body, and nitrogen–oxygen mixture (air) as the filling gas are chosen. The problem was solved in a three-dimensional statement.

The two-dimensional laminar flow of a viscous incompressible fluid over a flat surface is considered at high Reynolds numbers. The influence exerted on the Blasius boundary layer by a body moving downstream with a low velocity relative to the plate is studied within the framework of asymptotic theory. The case in which a small external body modeled by a potential dipole moves downstream at a constant velocity is investigated. Formally, this classical problem is nonstationary, but, after passing to a coordinate system comoving with the dipole, it is described by stationary solutions of boundary layer equations on the wall moving upstream. The numerically found solutions of this problem involve closed and open separation zones in the flow field. Nonlinear regimes of the influence exerted by the dipole on the boundary layer with counterflows are calculated. It is found that, as the dipole intensity grows, the dipole-induced pressure acting on the boundary layer grows as well, which, after reaching a certain critical dipole intensity, gives rise to a singularity in the flow field. The asymptotics of the solution near the isolated singular point of the flow field is studied. It is found that, at this point, the vertical velocity grows to infinity, viscous stress vanishes, and no solution of the problem exists at higher dipole intensities.

The penalized wall function method for simulation of compressible near-wall turbulent flow regions in the numerical modeling of viscous compressible flows is developed. The method is formulated as a differential condition to match the outer and the wall function solutions and is based on a generalized characteristic-based volume penalization method to transfer shear stress from the outer region of the boundary layer to the wall. The method is modified to extend its applicability to turbulent flows with adverse pressure gradient, when separation and reattachment zones are formed, as well as to use computational meshes with coarser near-wall resolution. These advantages are demonstrated for two test problems, namely, the flow over a flat plate with zero and adverse pressure gradients.

A survey of works concerning difficulties associated with harmonic grid generation in plane domains with angles and cutouts is given, and some new results are presented. It is well known that harmonic grids produced by standard methods in domains with cutouts or reentrant angles (i.e., interior angles greater than π) may contain defects, such as self-overlappings or exit beyond the domain boundary. It is established that, near the vertex of a reentrant angle, these defects follow from the asymptotics constructed for the underlying harmonic mapping, according to which the grid line leaving the angle vertex is tangent to one of the angle sides at the vertex (an effect referred to as “adhesion”), except for a special case. A survey of results is given for domains z of three types with angles or cutouts (L-shaped, horseshoe, and a domain with a rectangular cutout), for which standard methods for harmonic grid generation encounter difficulties. Applying the multipole method to such domains yields a harmonic mapping for them with high accuracy: the a posteriori error estimate of the mapping in the C(z)  norm is 10–7 in the case of using 120 approximative functions.

For the problem of conformal mapping of a half-plane onto a Z-shaped domain with arbitrary geometry, an efficient method is developed for finding parameters of the Schwarz–Christoffel integral, i.e., the preimages of the vertices (prevertices) and the pre-integral multiplier. Special attention is given to the situation of crowding prevertices, in which case conventional integration methods face significant difficulties. For this purpose, the concept of a cluster is introduced, its center is determined, and all integrand binomials with prevertices from this cluster are expanded into a fast-convergent series by applying a unified scheme. Next, the arising integrals are reduced to single or double series in terms of Gauss hypergeometric functions F(a, b, c, q). The fast convergence of the resulting expansions is ensured by applying formulas for analytic continuation of  F(a, b, c, q) to a neighborhood of the point q = 1 and using numerically stable recurrence relations. The constructed expansions are also fairly efficient for choosing initial approximations for prevertices in Newton’s iteration method. By using the leading terms of these expansions, the approximations for the prevertices are expressed in explicit form in terms of elementary functions, and the subsequent iterations ensure the fast convergence of the algorithm. After finding the parameters in the integral, the desired mapping is constructed as a combination of power series expansions at prevertices, regular expansions at the preimage of the center of symmetry, a Laurent series in a semi-annulus, and special series near the preimages of the vertical segments. Numerical results demonstrate the high efficiency of the developed method, especially in the case of strong crowding of prevertices.

In this paper, we consider a black-box multiobjective optimization problem, whose objective functions are computationally expensive. We propose a density function-based trust region algorithm for approximating the Pareto front of this problem. At every iteration, we determine a trust region and then in this trust region, select several sample points, at which are evaluated objective function values. In order to obtain non-dominated solutions in the trust region, we convert given objective functions into one function: scalarization. Then, we construct quadratic models of this function and the objective functions. In current trust region, we find optimal solutions of all single-objective optimization problems with these models as objectives. After that, we remove dominated points from the set of obtained solutions. In order to estimate the distribution of non-dominated solutions, we introduce a density function. By using this density function, we obtain the most “isolated” point among the non-dominated points. Then, we construct a new trust region around this point and repeat the algorithm. We prove convergence of proposed algorithm under the several assumptions. Numerical results show that even in case of tri-objective optimization problems, the points generated by proposed algorithm are uniformly distributed over the Pareto front.

This paper purposes to present a new discrete scheme for the singularly perturbed semilinear Volterra–Fredholm integro-differential equation including two integral boundary conditions. Initially, some analytical properties of the solution are given. Then, using the composite numerical integration formulas and implicit difference rules, the finite difference scheme is established on a uniform mesh. Error approximations for the approximate solution and stability bounds are investigated in the discrete maximum norm. Finally, a numerical example is solved to show -uniform convergence of the suggested difference scheme.

We construct high order local discontinuous Galerkin (LDG) discretizations coupled with third and fourth order backward differentiation formulas (BDF) for the Allen–Cahn equation. The numerical discretizations capture the advantages of linearity and high order accuracy in both space and time. We analyze the stability and error estimates of the time third-order and fourth-order BDF-LDG discretizations for numerically solving Allen–Cahn equation respectively. Theoretical analysis shows the stability and the optimal error results of theses numerical discretizations, in the sense that the time step τ requires only a positive upper bound and is independent of the mesh size h. A series of numerical examples show the correctness of the theoretical analysis. Comparison with the first-order numerical discretization illustrates that the high order BDF-LDG discretizations show good performance in solving stiff problems.