Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
ISSN (print): 0044-4669
Founders: Russian Academy of Sciences, Federal Research Center IU named after. A. A. Dorodnitsyna RAS
Editor-in-Chief: Evgeniy Evgenievich Tyrtyshnikov, Academician of the Russian Academy of Sciences, Doctor of Physics and Mathematics sciences, professor
Frequency / access: 12 issues per year / Subscription
Included in: White List (2nd level), Higher Attestation Commission list, RISC, Mathnet.ru
Media registration certificate: № 0110141 от 04.02.1993
Current Issue
Vol 63, No 12 (2023)
ЮБИЛЕЙ
МАТЕМАТИЧЕСКАЯ ФИЗИКА
Numerical Analysis of Rarefied Gas Flow through a System of Short Channels
Abstract
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.
Evolution of the Shape of a Gas Cloud during Pulsed Laser Evaporation into Vacuum: Direct Simulation Monte Carlo and the Solution of a Model Equation
Abstract
The dynamics of gas expansion during nanosecond laser evaporation into vacuum is studied. The problem is considered in an axisymmetric formulation for a wide range of parameters: the number of evaporated monolayers and the size of the evaporation spot. To obtain a reliable numerical solution, two different kinetic approaches are used—the direct simulation Monte Carlo method and solution of the BGK model kinetic equation. The change in the shape of the cloud of evaporated substance during the expansion process is analyzed. The strong influence of the degree of rarefaction on the shape of the forming cloud is shown. When a large number of monolayers evaporate, good agreement with the continuum solution is observed.
Numerical and Theoretical Analysis of Model Equations for Multicomponent Rarefied Gas
Abstract
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.
On the Simulation of a Rarefied Plasma Jet on the Basis of Kinetic Equations
Abstract
The problem of a rarefied plasma jet emerging from a stationary plasma engine is considered. The consideration is carried out entirely at the kinetic level; namely, the motion of all plasma components is described in terms of distribution functions. The system of kinetic equations should be solved together with Maxwell’s equations. Methods for solving the resulting problem are discussed.
Data Parallelization Algorithms for the Direct Simulation Monte Carlo Method for Rarefied Gas Flows on the Basis of OpenMP Technology
Abstract
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.
Nonclassical Heat Transfer in a Microchannel and a Problem for Lattice Boltzmann Equations
Abstract
A one-dimensional problem of heat transfer in a bounded domain (microchannel) filled with rarefied gas is considered. Two molecular beams enter the domain from the left boundary, the velocities of the particles are equal in the each beam. The diffuse reflection condition is set on the right boundary. It is shown using the Shakhov kinetic model that by varying the ratio of velocities in the molecular beams it is possible to obtain a heat flux of various magnitudes and signs such that the te-mperatures on the left and right boundaries are equal or the temperature gradient in the boundary layer has the same sign as the heat flux. This problem is related to the problem of constructing lattice Boltzmann equations with four velocities, which can reproduce the first Maxwell half-moments. It is shown that in this case the optimal ratio of discrete velocities is 1 : 4.
Study of Nonclassical Transport by Applying Numerical Methods for Solving the Boltzmann Equation
Abstract
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.
Accelerating the Solution of the Boltzmann Equation by Controlling Contributions to the Collision Integral
Abstract
A method of reducing the number of arithmetic operations needed to evaluate the Boltzmann collision integral by the conservative projection method is proposed. This is achieved by eliminating the contributions that are less than a certain threshold. An estimate of the maximum magnitude of this threshold is given. For four such thresholds that differ by an order of magnitude from each other, calculations of the flows of rarefied gas at Mach numbers in the range from 0.5 to 10 are carried out, and the results are compared with those obtained using the basic method. In all cases, there is a slight (within a few percent) difference for the highest threshold and almost complete coincidence for the other thresholds. A multiple acceleration of the solution of the Boltzmann equation was obtained, which is most significant for large Mach numbers.
Three-Dimensional Simulation of a High-Velocity Body Motion in a Tube with Rarefied Gas
Abstract
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.
On One Method for Calculating Nonstationary Heat Transfer between a Gas Flow and a Solid Body
Abstract
A method for calculating the nonstationary thermal interaction between a viscous gas flow and a solid body is presented. The method consists in direct joint integration over time of the equations of gas dynamics of a multicomponent mixture and the heat equation in a solid on multi-block unstructured meshes. To calculate one time step, the system of governing equations is split into hyperbolic and parabolic subsystems. The numerical method provides approximation of the matching condition (continuity of temperature and the normal component of the heat flux) at the interface between gas and solid and is efficient for nonstationary calculations. The comparison with the analytical solution of the model problem of the interaction of a high-speed flow and a heated plate confirm the efficiency of the method.
Singularity Formation in an Incompressible Boundary Layer on an Upstream Moving Wall under Given External Pressure
Abstract
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.
Stability Analysis of Polymerization Fronts
Abstract
Анализ устойчивости фронтов полимеризации.
В статье исследуется влияние некоторых параметров на условия устойчивости фронта реакции в жидкой среде. Математическая модель состоит из уравнения теплопроводности, уравнения концентрации и уравнения Навье–Стокса в приближении Буссинеска. Асимптотический анализ проводился с использованием приближения, предложенного Зельдовичем и Франк-Каментским для решения проблемы интерфейса. Анализ стабильности был проведен для получения линеаризованной задачи, которая будет решаться численно с использованием мультиквадратного радиального базиса методом функции для нахождения конвективного порога. Это позволит сделать вывод о влиянии каждого параметра на стабильность фронта, в частности амплитуду и резонансную частоту.
Generalisation of the Penalised Wall Function Method for the Simulation of Turbulent flows With Unfavourable Pressure Gradients
Abstract
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.
ОБЩИЕ ЧИСЛЕННЫЕ МЕТОДЫ
Analysis of Defects and Harmonic Grid Generation in Domains with Angles and Cutouts
Abstract
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.
Projection-Grid Schemes on Irregular Grids for a Parabolic Equation
Abstract
A family of projection-grid schemes has been constructed for approximating parabolic equations with a variable diffusion coefficient in tensor form. The schemes are conservative and retain the self-adjointness of the original differential operator and are destined for calculations on 3D irregular difference grids, including tetrahedral, mixed (grids of arbitrary polyhedra), and locally adaptive (octal-tree type).
Conformal Mapping of a Z-Shaped Domain
Abstract
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.
Study of the Gardner Equation with Homogeneous Boundary Conditions Via Fourth Order Modified Cubic B-Spline Collocation Method
Abstract
Исследование уравнения Гарднера с однородными граничными условиями с помощью метода коллокаций с кубическими B-сплайнами для модифицированного уравнения четвертого порядка.
Исследуется уравнение Гарднера, которое преобразуется в связанную систему нелинейных дифференциальных уравнений в частных производных, и для нахождения его численного решения применяется модифицированный метод коллокации кубических B-сплайнов. Дискретизация по времени и линеаризация уравнения Гарднера были выполнены с использованием метода Кранка–Николсона и соответствующей квазилинеаризации. Получена система линейных алгебраических уравнений, анализ которой по методу Неймана показал условную устойчивость. Численные исследования этого уравнения проведены в различных постановках, таких как распространение начального положительного импульса и волны с изломом, распространение и взаимодействие двух солитонов, образование волны из одного солитона, эволюция стоячих солитонов. Полученные результаты сравнены с имеющимися в литературе и наиболее достоверными. Вычисляются также интегралы в этой задаче, чтобы показать справедливость законов сохранения. Численные результаты демонстрируют высокую точность и обоснованность настоящего метода.
ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ
Density Function-Based Trust Region Algorithm for Approximating Pareto Front of Black-Box Multiobjective Optimization Problems
Abstract
Основанный на функции плотности алгоритм определения доверительной области для аппроксимации границы Парето задач многоцелевой оптимизации типа “черный ящик”.
Рассматривается задача многоцелевой оптимизации типа “черный ящик”, целевые функции которой требуют больших вычислительных затрат. Предложен основанный на функции плотности алгоритм оценки доверительной области для аппроксимации границы Парето этой задачи. На последовательных итерациях определяется граница доверительной области, а затем в ней выбирается несколько точек выборки, в которых оцениваются значения целевой функции. Для получения в такой области решения без доминирования заданные целевые функции преобразуются в одну скалярную функцию. Затем строятся модели с квадратичным характером целевых функций. В текущей доверительной области находятся решения всех задач оптимизации с одной целевой функцией. Затем удаляются доминирующие точки из множества полученных решений. Для оценки распределения решений без доминирования вводится функция плотности, используя которую получены наиболее “изолированные” точки. Доказана сходимость предложенного алгоритма при некоторых допущениях. Численные результаты показывают, что, даже в случае задач оптимизации с тремя целями, точки, генерируемые предложенным алгоритмом, равномерно распределяются по границе Парето.
ОБЫКНОВЕННЫЕ ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ
A Uniformly Convergent Numerical Method for Singularly Perturbed Semilinear Integro-Differential Equations with Two Integral Boundary Conditions
Abstract
Равномерно сходящийся численный метод для решения сингулярно возмущенных полулинейных интегродифференциальных уравнений с двумя интегральными граничными условиями.
Целью данной статьи является представление новой дискретной схемы для сингулярно возмущенной полулинейной системы. Интегродифференциальное уравнение Вольтерра–Фредгольма включает два интегральных граничных условия. Приведены некоторые основные аналитические свойства решения, а затем с помощью составных формул численного интегрирования построена неявная разностная схема на равномерной сетке. Дана оценка погрешности приближенного решения и приведены границы устойчивости в дискретной равномерной норме. Представлен численный пример, иллюстрирующий е-равномерную сходимость предложенной разностной схемы.
A Novel Fitted Approach for the Solution of a Class of Singularly Perturbed Differential-Difference Equations Involving Small Delay in Undifferentiated Term
Abstract
Новый адаптивный метод решения для класса сингулярно возмущенных дифференциально-разностных уравнений с малой сдвижкой в недифференциальном члене.
Рассмотрен метод решения класса сингулярно возмущенных дифференциально-разностных уравнений с малой сдвижкой. С помощью разложения в ряд Тейлора задача сводится к эквивалентной версии исходной задачи, для которой затем предложена новая трехчленная конечно-разностная рекуррентная схема ее решения. Неоднородность решения преодолевается введением подходящего параметра настройки в полученной схеме. Итоговая система алгебраических уравнений решается с помощью дискретно-инвариантного алгоритма. Исследованы устойчивость и сходимость метода и дано приложение этого подхода к решению нескольких тестовых задач. Приведенные примеры показывают, что метод способен хорошо аппроксимировать решение со скоростью сходимости второго порядка.
УРАВНЕНИЯ В ЧАСТНЫХ ПРОИЗВОДНЫХ
Stability and Error Estimates of High Order Bdf-Ldg-Discretizations for the Allen–Cahn Equation
Abstract
Устойчивость и оценки погрешности метода Галеркина высокого порядка для уравнения Аллена–Кана.
Исследовано применение метода Галеркина высокого порядка с локальными разрывами в сочетании с формулами дифференцирования против потока третьего и четвертого порядков для уравнения Аллена–Кана. Численная дискретизация обеспечивает преимущества линейности и высокой точности как по пространству, так и по времени. Проанализированы оценки устойчивости и погрешности дискретизации по времени третьего порядка и четвертого порядка в приложении к численному решению уравнения Аллена–Кана. Теоретический анализ показывает устойчивость и оптимальные результаты погрешности этих численных дискретизаций в том смысле, что шаг по времени должен быть положительным и при этом он не зависит от шага сетки. Ряд численных примеров показал справедливость проведенного анализа. Сравнение с численной дискретизацией первого порядка показывает, что предложенная дискретизация высокого порядка имеет высокую эффективность при решении жестких задач.
Multipole Representation of the Gravitational Field for Asteroid (16) Psyche
Abstract
An approach to calculating multipole approximations of the gravitational potential of small celestial bodies with an irregular mass distribution is demonstrated for the asteroid (16) Psyche as an example.