The motion and ablation of a meteoroid breaking up into a large number of fragments are considered. At the first stage, the fragments move with a common shock wave, before dispersing to a distance sufficient for independent motion. We consider models of cloud of fragments that simulate the meteoroid disruption at this stage: the two-parameter model, which takes into account changes in the cloud shape and density, and simple models used in the literature that do not take these effects into account. The models differ in the equations for the lateral expansion rate of the cloud. The unrealistically strong increase in the midsection radius, which is given by simple models, is usually limited in the literature to a certain specified value. The effect of this midsection radius cutoff in different fragment cloud models on the results of modeling the energy deposition of the Chelyabinsk superbolide is studied. For this purpose, the equations of the physical theory of meteors are solved numerically using the same ablation model developed by the authors for different fragmentation models. The influence of the heat transfer coefficient on the energy deposition of the bolide obtained using different fragment cloud models and the applicability of these models are studied.

A bicompact scheme for the Navier–Stokes equations is considered in the case of a compressible heat-conducting fluid. The scheme is constructed using splitting by physical processes and it has the fourth order of approximation in space and the second order of approximation in time. New, conservative formulas are derived for transitions between two different representations of the numerical solution in bicompact schemes for hyperbolic and parabolic equations. The parallel implementation of the bicompact scheme is tested for strong scalability. The bicompact scheme is applied to the three-dimensional direct numerical simulation of the mixing layer with convective Mach numbers of 0.4 and 0.8. In the calculated flows, the zone of turbulent mixing is resolved in detail, and the phenomena observed in experiments are adequately reproduced. Good quantitative agreement is demonstrated with the simulations carried out by other authors.

This paper presents results of computational and experimental studies of the evolution of turbulent mixing in three-layer gas systems with the development of hydrodynamic instabilities, in particular, the Richtmyer–Meshkov and Kelvin–Helmholtz instabilities, under the action of shock waves. One of the contact boundaries between gases is flat, while the other one has the form of a chevron. The numerical simulations are carried out both with and without initial perturbations of contact boundaries. It is shown that the roughness of the contact boundary significantly affects the width of the mixing zone.

Upwind bicompact schemes of third-order approximation in space are presented for the first time. A formula is obtained for the transition factor of an arbitrary fully discrete bicompact scheme with Runge–Kutta time stepping. Stability and monotonicity of a scheme of first order in time are investigated, and the dissipative and dispersion properties of a scheme of third order in time are analyzed. Advantages of the new schemes over their centered counterparts are demonstrated.

Рассматривается бикомпактная схема для уравнений Навье-Стокса в случае сжимаемой теплопроводной жидкости. Схема строится методом расщепления по физическим процессам, имеет четвертый порядок аппроксимации по пространству и второй -- по времени. Выводятся новые, сохраняющие консервативность формулы перехода между двумя различными представлениями численного решения в бикомпактных схемах для уравнений гиперболического и параболического типов. Параллельная программная реализация бикомпактной схемы тестируется на сильную масштабируемость по числу процессов. По бикомпактной схеме проводится трехмерное прямое численное моделирование слоя смешения при конвективных числах Маха 0.4 и 0.8. В рассчитанных течениях детально разрешается зона турбулентного перемешивания, адекватно воспроизводятся явления, наблюдаемые в эксперименте. Демонстрируется хорошее количественное согласие с моделированием, проведенным другими авторами.

A new test problem for one-dimensional gas dynamics equations is considered. Initial data in the problem is a periodic smooth wave. Shock waves are formed in the gas flow over a finite time. The convergence under mesh refinement is analyzed for two third-order accurate linear schemes, namely, a bicompact scheme and Rusanov’s scheme. It is demonstrated that both schemes have only the first order of integral convergence in the shock influence area. However, when applied to equations of isentropic gas dynamics, the schemes converge with at least the second order.

Various approaches are developed for modeling the destruction of cosmic bodies in the atmosphere into a large number of fragments. It is assumed that, at first, they move with a common shock wave as a single body deformed under the action of pressure forces. Four models of meteoroid fragmentation into a cloud of fragments are considered: two developed by us and two that are generally accepted. The main distinctions between the models are revealed. These models are used to obtain numerical solutions of the equations of meteor physics taking into account ablation to calculate the interaction of the Chelyabinsk meteoroid with the atmosphere. Solutions for different models are compared with each other and with observational data. After the fragments diverge over a large distance, their independent motion is considered. For the probabilistic and cumulative mass distribution of fragments, formulas are obtained using the results of experiments on the destruction of bodies under high-speed impact. The cumulative distribution is compared with the experimental data and with the mass distributions of recovered meteorites in the cases of a significant number of found fragments. The mass of the fragmented meteoroid and the luminosity are found by integrating over all fragments using their probabilistic mass distribution as the initial one. The mass and velocity of each fragment are determined from the equations of meteor physics.

For the first time, bicompact schemes have been generalized to nonstationary Navier–Stokes equations for a compressible heat-conducting fluid. The proposed schemes have an approximation of the fourth order in space and the second order in time, and they are absolutely stable (in the frozen-coefficients sense), conservative, and efficient. One of the new schemes is tested on several two-dimensional problems. It is shown that when the mesh is refined, the scheme converges with an increased third order. A comparison is made with the WENO5-MR scheme. The superiority of the chosen bicompact scheme in resolving vortices and shock waves, as well as their interaction, is demonstrated.

This paper presents the experimental and numerical results of studying the growth dynamics of the deterministic and given initial perturbations defined in a certain way. The formation, growth, and further evolution of inhomogeneities of the contact boundary occurs due to the development of the Rayleigh–Taylor instability (RTI) at the gas-liquid interface, and in particular (in this study), the air-water interface. The significant difference in the densities of the selected substances leads to a noticeable slowdown in the dynamics of the Kelvin–Helmholtz instability (KHI), which is responsible for the formation of mushroom-like structures, and, as a result, to the longer growth of water jets and the later moment of their destruction and transition to mixing. In this study, a quantitative comparison of the physical data recorded on the original experimental setup, which is described in this paper, with the calculated data obtained using various numerical methods is carried out. The numerical modeling is based on a complete 2D hydrodynamic model for describing the dynamics of the development of the RTI. The surface tension (water-air) and viscosity (water or air) are neglected in this study. The parameters of the development of the instability measured in the experiment and found in the calculations indicate satisfactory agreement between the obtained data. The quantitative results presented in this study justify the use of the classical hydrodynamics model to describe the movements of liquid and gas observed in this experiment and the fairly accurate numerical implementation of the corresponding model in the difference methods used here. The investigation of the development of turbulent mixing depending on well-defined initial conditions and the new regularities of the laws of mixing of the media of different densities that arise in this case is an important element in the study.

High-order bicompact schemes for hyperbolic systems of conservation laws are considered. We aim to significantly speed up these schemes. Implicit-explicit Runge–Kutta methods are proposed for time discretization, instead of the previously used diagonally implicit methods. The global Lax–Friedrichs–Rusanov flux splitting is a premise for the implicit-explicit approximation. It is shown that implicit-explicit bicompact schemes are stable for any ratio of steps in time and space. The accuracy of the new implicit-explicit schemes and the substantial speed-up achieved are demonstrated on multidimensional gas dynamics problems.

For the three-dimensional Euler equations, a locally one-dimensional bicompact scheme having the fourth order of approximation in space and the second order of approximation in time is considered. The scheme is used in the Taylor–Green vortex problem in an inviscid perfect gas to examine the degree to which a conservative limiting (monotonization) method applied to bicompact schemes affects their theoretically high spectral resolution. Two parallel computational algorithms for locally one-dimensional bicompact schemes are proposed. One of them is used for carrying out computations. It is shown that, in the case of monotonization, the chosen bicompact scheme resolves 70–85% of the kinetic energy spectrum of the fluid. The scheme is compared with high-order accurate WENO5 schemes in terms of the behavior of kinetic energy and enstrophy. It is demonstrated that the bicompact scheme has noticeably lower dissipation and more weakly suppresses medium-scale eddies.

A survey of works concerning high-order accurate numerical methods designed for shock-capturing computations of discontinuous solutions to hyperbolic systems of conservation laws is presented. The basic problems arising in the theory of such methods are formulated, and approaches to their solution are proposed. Primary attention is given to fundamentally new shock-capturing methods (known as combined schemes) that monotonically localize shock fronts, while preserving high accuracy in shock influence areas. Test computations are presented that demonstrate the significant advantages of combined schemes over standard NFC ones when applied to computing discontinuous solutions with shock waves.

В настоящей работе представлены экспериментальные и численные результаты исследования динамики роста детерминированных, определенным образом заданных начальных возмущений. Возникновение, рост и дальнейшая эволюция неоднородностей контактной границы происходит благодаря развитию неустойчивости Рэлея-Тейлора на границе раздела газ-жидкость, в частности (в данной работе), воздухвода. Существенная разница плотностей выбранных веществ приводит к заметному замедлению динамики неустойчивости Кельвина-Гельмгольца, отвечающей за образование грибообразных структур, и, как следствие, к более длительному росту струй воды и более позднему моменту начала их разрушения и перехода к перемешиванию. Выполнено количественное сопоставление натурных данных, зафиксированных на оригинальной экспериментальной установке, описание которой приводится в настоящей работе, с расчетными данными, полученными с использованием различных численных методик. В основе численного моделирования лежит полная 2D гидродинамическая модель описания динамики развития неустойчивости Рэлея-Тейлора. Поверхностным натяжением (вода-воздух) и вязкостью (воды или воздуха) в данном исследовании пренебрегается. Измеренные в эксперименте и найденные в расчетах параметры развития неустойчивости свидетельствует об удовлетворительном согласии полученных данных. Приведенные в данном исследовании количественные результаты оправдывают использование модели классической гидродинамики для описания наблюдаемых в данном опыте движений жидкости и газа и достаточно точную численную реализацию соответствующей модели в применяемых здесь разностных методиках. Существенным элементом проведенного исследования является изучение развития турбулентного перемешивания в зависимости от вполне определенных начальных условий и возникающих в этом случае новых закономерностей законов перемешивания разноплотных сред.

High-order bicompact schemes are designed for the quasilinear multidimensional diffusion equation. They are constructed using the method of lines, the finite volume method, the bicubic (or tricubic) Hermite interpolation. Implicit-explicit and diagonally-implicit Runge-Kutta methods are applied to time integration. The resulting schemes are stable for any ratios between grid steps, are conservative, have approximation of the fourth order in space and the third order in time. To implement the implicit-explicit schemes, an iteration method is proposed based on the approximate factorization of their multidimensional difference operators. This method is modified to implement the schemes with diagonally-implicit Runge-Kutta time stepping. High-order grid convergence and implementation efficiency of the new bicompact schemes are demonstrated on numerical examples.

Рассматриваются высокоточные бикомпактные схемы для гиперболических систем законов сохранения. Ставится цель значительно повысить скорость счета по этим схемам. Предлагается применять для дискретизации по времени неявно-явные методы Рунге-Кутты, вместо использовавшихся ранее диагонально-неявных методов. Предпосылкой для неявно-явной аппроксимации оказывается глобальное потоковое расщепление Лакса-Фридрихса-Русанова. Показывается, что неявно-явные бикомпактные схемы устойчивы при любых соотношениях шагов по времени и пространству. На ряде многомерных задач газовой динамики демонстрируется высокая точность новых неявно-явных схем и многократное ускорение счета.

Bicompact schemes are generalized for the first time to the linear multidimensional convection–diffusion equation. Schemes are constructed using the method of lines, the finite-volume method, and bi- and tricubic Hermite interpolation of the sought function in a cell. Time stepping is based on diagonally implicit Runge–Kutta methods. The proposed bicompact schemes are unconditionally stable, conservative, and fourth-order accurate in space for sufficiently smooth solutions. The constructed schemes are implemented by applying an efficient iterative method based on approximate factorization of their multidimensional equations. Every iteration of the method is reduced to a set of independent one-dimensional scalar two-point Gaussian eliminations. Several stationary and nonstationary exact solutions are used to demonstrate the high-order convergence of the developed schemes and the fast convergence of their iterative implementation. Advantages of bicompact schemes as compared with Galerkin-type finite-element schemes are discussed.

Fully discrete bicompact schemes of the fourth order of approximation in space are investigated for entropy stability in problems of gas dynamics. Expressions for the rate of entropy production in these schemes are derived. Qualitative estimates are obtained for the behavior of this quantity. On the example of one-dimensional Riemann test problems, a numerical analysis of the entropy production rate for bicompact schemes of the first and third orders of approximation in time is carried out. From the results of this analysis, it is concluded whether any entropy correction is necessary for bicompact schemes.

Построена энтропийная регуляризация консервативного устойчивого разрывного метода Галеркина в консервативных переменных для двумерных уравнений Эйлера на основе специального ограничителя наклонов. Данный ограничитель обеспечивает выполнение двумерных аналогов условий монотонности и дискретного аналога энтропийного неравенства. Проведено тестирование разработанного метода на двумерных модельных газодинамических задачах.