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The paper gives a review of using the semi-Lagrangian approximation depending on the fulfillment of conservation laws for the transfer operator. We begin with approximations of the one-dimensional transfer equation and a parabolic one as simple methodological examples. For two-dimensional problems, first we apply one-dimensional approximations in two directions separately. Then we present another combined approximation along trajectories of the transfer operator. For parabolic and transfer equations, the principles of constructing discrete analogues are demonstrated for three different conservation laws of transfer operator (or the requirements of stability in the related discrete norms similar to the L1, L2, L∞ - norms). It is significant that different conservation laws yield distinct difference problems as well as distinct ways to justify their stability. [ABSTRACT FROM AUTHOR]

In the paper, a semi-Lagrangian approximation is presented for the numerical solution of the two-dimensional timedependent Navier-Stokes equations for viscous heat-conducting gas. In each equation, a combination of three first-order derivatives describing the transfer of a corresponding substance (density, velocity components, or internal energy) along trajectories is interpreted as the "transfer derivative" in the transfer direction. The other terms of the equations are written in the Euler form. On the sought-for time level, the standard conforming finite element method is realized for them with the linear elements on triangles and the bilinear ones on rectangles. The stencil adaptation along trajectories enables us to avoid the Courant-Friedrichs-Lewy upper limit which describes the dependence of the time step on the mesh-size of the space triangulation. At the end of the paper, a numerical example illustrates the implementation of the described algorithms. [ABSTRACT FROM AUTHOR]

In the paper, a mathematical model and a numerical algorithm are proposed for modeling the complex of phenomena which accompany the passage of a friable asteroid-comet body through the Earth's atmosphere: the material ablation, the dissociation of molecules, and the radiation. The proposed model is constructed on the basis of the Navier-Stokes equations for viscous heat-conducting gas with an additional equation for the motion and propagation of a friable lumpy-dust material in air. The energy equation is modified for the relation between two its kinds: the usual energy of the translation of molecules (which defines the temperature and pressure) and the combined energy of their rotation, oscillation, electronic excitation, dissociation, and radiation. For the mathematical model of atmosphere, the distribution of density, pressure, and temperature in height is taken as for the standard atmosphere. An asteroid-comet body is taken initially as a round body consisting of a friable lumpy-dust material with corresponding density and significant viscosity which far exceed those for the atmosphere gas. A numerical algorithm is proposed for solving the initial-boundary problem for the extended system of Navier-Stokes equations. The algorithm is the combination of the semi-Lagrangian approximation for Lagrange transport derivatives and the conforming finite element method for other terms. The implementation of these approaches is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

In the paper, a semi-Lagrangian approximation is presented for the numerical solution of the two-dimensional timedependent Navier-Stokes equations for viscous heat-conducting gas. In each equation, a combination of three first-order derivatives describing the transfer of a corresponding substance (density, velocity components, or internal energy) along trajectories is interpreted as the "transfer derivative" in the transfer direction. The other terms of the equations are written in the Euler form. On the sought-for time level, the standard conforming finite element method is realized for them with the linear elements on triangles and the bilinear ones on rectangles. The stencil adaptation along trajectories enables us to avoid the Courant-Friedrichs-Lewy upper limit which describes the dependence of the time step on the mesh-size of the space triangulation. At the end of the paper, a numerical example illustrates the implementation of the described algorithms. [ABSTRACT FROM AUTHOR]

In the paper, a mathematical model and a numerical algorithm are proposed for modeling the complex of phenomena which accompany the passage of a friable asteroid-comet body through the Earth's atmosphere: the material ablation, the dissociation of molecules, and the radiation. The proposed model is constructed on the basis of the Navier-Stokes equations for viscous heat-conducting gas with an additional equation for the motion and propagation of a friable lumpy-dust material in air. The energy equation is modified for the relation between two its kinds: the usual energy of the translation of molecules (which defines the temperature and pressure) and the combined energy of their rotation, oscillation, electronic excitation, dissociation, and radiation. For the mathematical model of atmosphere, the distribution of density, pressure, and temperature in height is taken as for the standard atmosphere. An asteroid-comet body is taken initially as a round body consisting of a friable lumpy-dust material with corresponding density and significant viscosity which far exceed those for the atmosphere gas. A numerical algorithm is proposed for solving the initial-boundary problem for the extended system of Navier-Stokes equations. The algorithm is the combination of the semi-Lagrangian approximation for Lagrange transport derivatives and the conforming finite element method for other terms. The implementation of these approaches is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]