A CLASS OF EXACT SOLUTIONS FOR A SYSTEM OF RADIATIVE HEAT TRANSFER EQUATIONS

B. P. Tikhomirov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 3-8.

      A spectral formulation and a grey matter problem are used to derive an exact solution for a nonequilibrium heat wave propagating with a constant velocity across a cold (nontransparent) material. Absorption, radiation, and isotropic scattering are accounted. Absorption and scattering coefficients are chosen in a special way. The system of energy and radiation transport equations and that of diffusion equations are considered both with and without an energy source (sink), depending on temperature. The exact solutions obtained may be used for numerical method improvements and program debugging.

V. I. Legonkov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 9-16.

      The paper discusses central notions involved in a modular/structured approach to large program development, that is the module notion and that of support system for modular/structured programming. A detailed definition of a module is given which is the main design element of an application program, its major properties and components are described. A general scheme for application program development with a modular/structured method is considered together with tool system components for supporting this scheme and their functions.

D. Yu. Anistratov, V. Ya. Goldin
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 17-23.

      A sphere-symmetry geometry was used to study a 1-D transport equation solved in a quasidiffusion form. The difference scheme considered is more accurate for critical parameter definitions and less work - intensive for iterations compared to the Vladimirov method, that of characteristic tubes and S_n - method, while being outperformed in accuracy by the quasidiffusion method, but the scheme considered is easier to implement and more cost-effective. We investigated Vladimirov - like schemes and a class of schemes for a 1-D transport equation in terms of their conservatism.

Yu. S. Bailcarov, L. V. Kalinichova, L. D. Parkhomenko, A. V. Suchkov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 24-28.

      The paper considers a general purpose interpretation-type open system for implementing data input, coutrol and update functions. The system includes a set of ES EVM Assembler and PL/1 programs.

N. V. Ivanov, L. Z. Morenko
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 29-36.

      The Monte-Carlo method is used to solve a transport equation. The photon path models use a modified majorizing cross-section technique which permits to avoid difficulties associated with tabulating pho- ton/medium interaction cross-sections. A majorizing cross-section, when properly chosen, yields a model efficiency at least of 0,5. Our results are in a good agreement with data reported by other authors.

A. S. Shvedov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 37-44.

      The paper cousiders an invariant difference scheme development for solving 3-D time-dependent equations in gas dynamics by decomposing the velocity vector into bases which include normal vectors and those along the main directions of coordinate surfaces. Equations for a compressible inviscid nonconductive gas behavior are presented in this way when only curvilinear velocity vector components are included and the differentiation over spatial variables is incorporated as some vector field divergences in these equations. After the transition to integral conservation laws and discretization have been accomplished we obtain an invarirnt difference scheme implemented as a computer program. The results of computations are given.

O. M. Kozyrev
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 45-49.

      An approach is described to establish exchange boundary conditions on a Lagrangian boundary for 2-D numerical integration of equations in gas dynamics. The exchange boundary conditions proposed are symmetric with respect to region numbering and do not lead to additional stability conditions compared to those for regions to be calculated.
      A Godunov scheme for 2-D acoustic equations on rectangular grids is used to describe exchange boundary conditions derivation and to justify the stability of associated difference complex problems. For the case of mesh nonconsistency on the boundary, a 2-D difference complex problem is proved to be stable using energy methods.

A. A. Sadovoj, N. M. Chulkov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 50-58.

      Inert ion-driven convergence of viscoplastic cylinder and sphere shells is investigated. An equation of motion for such finite-thickness shells is derived. The comparison made between kinetic-to-thermal energy dissipation rates shows that spheres have a higher dissipation rate than cylinders. The contribution of viscous and plastic properties to the kinetic energy dissipation is illustrated. Analitical and numerical results obtained may be used to analyze an inertion-driven convergence of shells and also to test complicated numerical procedures.

V. Ya. Goldin, D. A. Goldina, A.V. Kolpakov, A. V. Shilkov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 59-66.

      A system of equations in high temperature radiation gas dynamics is examined. We describe an algorithm for numerically solving 1-D problems in high - temperature radiation gas dynamics using a spherical geometry. An analysis is performed of an instability on the interface between two media due to a pulse flux transported by a nonequilibrium component, particularly, by a radiation. For models describing the radiation transport in terms of diffusion, the instability is shown to be avoided.

V. N. Pisarev
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 67-75.

      The paper studies a multiparameter family of ROMB schemes for a linear heat conduction equation. A parallel is drawn between the ROMB and three-point schemes. Equivalent three-point schemes are obtained both for temperatures and heat fluxes defined at mesh nodes. It is found that several well-known schemes belong to the parametric family of ROMB schemes. However, in some simplest cases the ROMB may be thought of as a weighted three-point scheme. Some issues concerning the underlying theory for this scheme are considered.

Yu. A. Bondarenko
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 76-79.

      The paper examines the stability of the difference D scheme for a 2-D gas dynamics in Lagrangian variables on arbitrary quad grids for an axisymmetric case where (as opposed to a plane case) the differential operators of a linearized difference scheme are not self-conjugate which may be a potential source of instability. The studies were carried out using a four-cell model which has a moving node shared by these cells. An example is given illustrating a mesh with an unsteady state of rest.

A. K. Gerasimenko, O. S. Shirokovskaya
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 80-83.

      We describe a cost-effective. a 1gorithm developed for determining the maximum eigenvalue of a transport operator. For methods which rely upon solving a stationary transport equation, such algorithms have a form of double iterations. There are various ways to perform iterations with respect to their duration: one may iterate "up to the end", that is until a desired accuracy is achieved, or more rough iterations are possible.
      The paper presents and compares two approaches for implementing iterations relying upon the above considerations in terms of their efficiency and cost-effectiveness. A more efficient and cost-effective algorithm is developed where the internal loop performs at most three iterations which allows to reduce the iteration number with a factor of 1,5-1,8 compared to the previous algorithm.

G. V. Dolgoleva, V. A. Zhmailo
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 84-89.

      We present a system of equations describing the flows in a rarefied and partially ionized gas. The equations were derived using two modified versions of the momentum method for solving a system of Boltzmann equations.
      The paper describes a difference scheme which approximates a source system of differential equations, a numerical method to solve it, and an example illustrating the method capabilities.

V. O. Lokutsievsky, O. V. Lokutsievsky
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1986. No 2. P. 90-94.

      A relatively cost-effective method is presented for obtaining approximate solutions of boundary value problems associated with parabolic equations. A class of applications is given where this method may be used. Our discussion is exemplified by the boundary value problem for a second-order equation. The method proposed is compared with others.