卷 58, 编号 8 (2018)

A linearized version of the classical Godunov scheme as applied to nonlinear discontinuity decays is described. It is experimentally shown that this version guarantees an entropy nondecrease, which makes it possible to simulate entropy growth on shock waves. The structure of shock waves after the discontinuity decays is studied. It is shown that the width of the shock waves and the time required for their formation depend on the choice of the Courant number. The accuracy of the discontinuous solutions is tested numerically.

A distinctive feature of hyperbolic equations is the finite propagation velocity of perturbations in the region of integration (wave processes) and the existence of characteristic manifolds: characteristic lines and surfaces (bounding the domains of dependence and influence of solutions). Another characteristic feature of equations and systems of hyperbolic equations is the appearance of discontinuous solutions in the nonlinear case even in the case of smooth (including analytic) boundary conditions: the so-called gradient catastrophe. In this paper, on the basis of the characteristic criterion for monotonicity, a universal algorithm is proposed for constructing high-order schemes monotone for arbitrary form of the sought-for solution, based on their analysis in the space of indefinite coefficients. The constructed high-order difference schemes are tested on the basis of the characteristic monotonicity criterion for nonlinear one-dimensional systems of hyperbolic equations.

A key difficulty faced when grid-characteristic methods on tetrahedral meshes are used to compute structures of complex geometry is the high computational cost of the problem. Formally, grid-characteristic methods can be used on any tetrahedral mesh. However, a direct generalization of these methods to tetrahedral meshes leads to a time step constraint similar to the Courant step for uniform rectangular grids. For computational domains of complex geometry, meshes nearly always contain very small or very flat tetrahedra. From a practical point of view, this leads to unreasonably small time steps (1–3 orders of magnitude smaller than actual structures) and, accordingly, to unreasonable growth of the amount of computations. In their classical works, A.S. Kholodov and K.M. Magomedov proposed a technique for designing grid-characteristic methods on unstructured meshes with the use of skewed stencils. Below, this technique is used to construct a numerical method that performs efficiently on tetrahedral meshes.

Two-dimensional conformal mappings are a powerful and elegant tool for solving many mathematical and physical problems. The conformal mapping method is suitable for constructing two-dimensional grids. The quasi-conformal mappings constructed in this paper naturally generalize the application of conformal mappings to grid construction in the three-dimensional case. For a steady irrotational flow of an ideal incompressible fluid, in addition to the velocity potential, two stream functions are introduced. Generalized Cauchy–Riemann conditions from which three-dimensional quasi-conformal mappings follow are presented. The mappings constructed can be represented as a sequence of two-dimensional conformal mappings. Examples of grid construction using the theory of quasi-conformal mappings are given. The best proof of these results is their visualization.

Based on numerical and analytical approaches, a comprehensive physical and mathematical model determining the motion and destruction of natural cosmic bodies in the Earth atmosphere is created. Multilevel interrelated problems are considered, including simulation of the aeroballistics of meteoroids and their fragments taking into account their thermal and mechanical destruction; computation of airflow around a set of bodies (the meteorite fragments); and analysis of the problem of multiple “explosions” in atmosphere, which is inherent in meteoroids moving in the atmosphere after their fragmentation. The aim of the study is to resolve an important aspect of the asteroid and comet threat—the interaction of meteoroids with the atmosphere.

A discrete mathematical model of hydrobiology of coastal zone is constructed and analyzed. The model takes into account the transport and transformation of polluting biogenic elements in water basins. The propagation and transformation of biogenic elements is affected by such physical factors as three-dimensional motion of water taking into account the advective transport and microturbulent diffusion, spatially inhomogeneous distribution of temperature, salinity, and oxygen. Biogenic pollutants typically arrive into the water basin with river flow, which depends on the weather and climate of the geographic region, or with drainage of insufficiently purified domestic and industrial waste or other kinds of anthropogenic impact. Biogenic pollutants can also appear due to secondary pollution processes, such as stirring up and transport of bed silt, shore abrasion, etc. Stoichiometric relations between biogenic nutrients for phytoplankton algae that can be used to determine the limiting nutrient for each species are obtained. Observation models describing the consumption, accumulation of nutrients, and growth of phytoplankton are considered. A three-dimensional mathematical model of transformation of forms of phosphorus, nitrogen, and silicon in the plankton dynamics problem in coastal systems is constructed and analyzed. This model takes into account the convective and diffusive transport, absorption, and release of nutrients by phytoplankton as well as transformation cycles of phosphorus, nitrogen, and silicon forms. Numerical methods for solving the problem that are based on high-order weighted finite difference schemes and take into account the degree of fill of the computation domain control cells are developed. These methods are implemented on a multiprocessing system. They make it possible to improve the accuracy of the numerical solution and decrease the computation time by several fold. Based on the numerical implementation, dangerous phenomena in coastal systems related to the propagation of pollutants, including oil spill, eutrophication, and algae bloom, which causes suffocation phenomena in water basins, are reconstructed.

The accuracy of the discontinuous Galerkin method of the third-order approximation on smooth solutions in the calculation of discontinuous solutions of a quasilinear hyperbolic system of conservation laws with shock waves propagating with a variable velocity is studied. As an example, the approximation of the system of conservation laws of shallow water theory is considered. On the example of this system, it is shown that, like the TVD and WENO schemes of increased order of approximation on smooth solutions, the discontinuous Galerkin method, despite its high accuracy on smooth solutions and in the localization of shock waves, reduces its order of convergence to the first order in the shock wave influence domain.

A multifluid mathematical model for the computation of a high speed collision of metallic plates is constructed. Each material—steel, of which the first plate is made; lead, of which the second plate is made; and the surrounding air—is assumed to be a compressible fluid. The Baer–Nunziato equations are solved. The determining system of equations is hyperbolic, and it is numerically solved using the HLL method. The problem statement corresponds to the full-scale experiment. A lead plate is thrown in the direction of a steel plate at a velocity of 500 m/s. Both plates have free boundaries. The main characteristics of the process—formation of shock waves, their propagation to the free boundaries of the plates, reflection in the form of rarefaction waves, and interaction of the rarefaction waves with the interface between the metals—are obtained in the computations. The relative error of the parameters of the shock waves compared with the known computational and experimental data does not exceed 7%. An estimate of the acceleration of the interface between the plates due to the passage of the rarefaction wave propagating from the free boundary of the steel plate is obtained.