USING THE GODUNOV SCHEME TO SOLVE SPHERICAL HARMONICS DIFFERENTIAL EQUATIONS FOR A 1-D KINETIC EQUATION

I. A. Adamskaya
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 3-8.

      The suitabi1ity of partial solutions orthogonal running (Godunov scheme) to spherical harmonics differential equations is justified for a 1-D stationary kinetic equation.
      It is proved that the problem boundary conditions permit to construct a complex system of linearly-independent vectors required to use the method considered.

Yu. A. Bondarenko, V. V. Zmushko, А. M. Stenin
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 9-12.

      The linearized difference scheme, "Crest", is used to describe a method for obtaining nonlocal and asymptotically exact stability conditions (with a mesh point number tending to infinity) from initial data in the presence of local inhomogeneity in the scheme coefficients. A case of region-by-region computation and that of one small cell, compared to the rest, are considered. Smeared shock wave stability is examined for an artificial quadratic viscosity.

V. I. Delov, L. V. Dmitrieva, A. M. Pastushenko, S. Yu. Suslova
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 13-18.

      A 1-D implicit unconditionally stable difference scheme is developed for solving 1-D numerical problems in gas dynamics. The differential equation discretization involves the same mesh distribution as for the classical "Crest" scheme using velocity values from semi-integral times. Finite-difference equations are solved by an iteration method which does not require a time step limitation similar to the Courant condition to converge. The performonce of the scheme developed is demonstrated for two test problems.

V. V. Bashurov, A. E. Ushakov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 19-24.

      A difference scheme for solving a transport equation with a nonlinear dissipative function (the so called D-viscosity) is examined. The difference equation is studied by the first difference approximation. Linearizing the first difference approximation in the vicinity of a partial solution gave uncertainty growth and its second derivative estimates. The artificial dissipative term (D-viscosity) is shown to correct the oscillations occrurring before they become fully grown and degrade the solution.

R. M. Shagaliev
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 25-33.

      For a 1-D case, a region-by-region algorithm is developed which approximates the conjugate conditions with 0(h² + τ) order. For some limiting cases, the unconditional algorithm stability is investigated and analytically proved. The algorithm is generalyzed to a 2-D case where the heat conduction equation is approximated with a node-type scheme. The numerical results indicate that the algorithm yields a high actual accuracy and unconditional stability for a 2-D heat conduction equation solved by regions on substantially nonorthogonal and nonuniform meshes with sufficiently large time steps.

A. Yu. Bisyarin, V. U. Gribov, A. D. Zubov, N. N. Pervinenko, V. E. Neuvazhaev, V. D. Frolov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 34-41.

      The paper treats the issues concerning a program package development for solving 2-D problems in computational physics for complicated geometry applications. This relies on reducing a complicated source problem to a sequence of simplified ones by distributing them according to geometry, physical processes, and space variables. Some aspects of deriving exchange boundary conditions are discussed for block computations. Design requirements are formulated for the programs included in a block-computation package. The paper gives a brief description of techniques implemented by application programs.

R. P. Tarasov, A. E. Temkin
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 42-48.

      Transport equations are used to examine the numerical simulation of a noncoherent optical radiation energy propagation through space-curved optical multimode waveguides. The transition from a wave equation to transport equations involves a fractional exponent of the Helmholtz operator and h^-1 pseudodifferential operator theory.

T. I. Zmushko, Yu. N. Bukharev
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 49-53.

      A method of space characteristics is used to study elastic wave propagation and forming the highest stress regions (that is maximum stress rate regions) for a thick sphere which experiences dynamic load on its external surface h = 0,5R, h is the shell thickness, R is the external radius). Die external load is assumed to be determined by two stress tensor components (σ_rr normal and τ_kθ tangent) and its application region varies with time. The sphere being studied occupies the R^l≤r≤R, region within the r, θ, φ spherical coordinates, where 0≤θ≤π/2, is uniform, isotropic, and has linearly elastic properties defined by the density, ρ, longitudinal wave velocity, α, and transverse wave velocity, β. Axial symmetry requirements are assumed to be met.

B. L. Voronin, O. M. Kozlova
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 54-57.

      A method is presented for scaling gas dynamic quantities due to the variation of a mesh which is used to solve 2-D numerical time-dependent problems in gas dynamics. The method peculiarity is that a 2-D mesh reconfiguration and gas dynamic quantities scaling reduce to progressively performing 1-D reconfiguration and scaling operations. The method capabilities are illustratied by a 2-D gas dynamics problem.

A. V. Lukshin, T. G. Sysoeva
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 58-61.

      A numerical solution dependence on the problem parameters is studied. A self-similar analytical solution is derived. The numerical solution obtained from an infinite column simulation is compared to a self-simiiar one. The isotherm is recovered using an approximate concentration front.

Yu. A. Bondarenko, A. M. Stenin
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 62-69.

      The paper describes a linearly approximated estimation of the Crest-type difference scheme stability for solving 2-D gas dynamics equations with Lagrangian variables on arbitrary rectangular meshes in terms of artificial viscosity in the case where a pressure for the first boundary part and a normal velocity component for the second are given. For a time step, sufficient stability conditions close to exact ones are obtained in the form of inequalities to efficiently account the cell shapes, the grid nonuniformity, viscosity and sound speed variability along with boundary condition types. A weak instability is revealed which depends on the grid configuration near the boundary and a scheme modification is given with such instability removed.

N. E. Balakirev, Yu. G. Bartenev, V. A. Erzunov, S. A. Zeldinova, N. A. Mustaeva, V. F. Tyurin
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 70-73.

      The adaptation of the DISPAK OS to the Elbrus-1 multiprocessor server is discussed. The reasons for creating a single OS for the BESM-6 and the server are examined. Adequate multiprocessor architecture mapping оп the operation system is analyzed.
      The DISPAK is the first opergtfon system implemented for a server. It provides the BESM-6 environment and the server compatibility, which permits the latter to use the software resources developed during the BESM-6 life time. However, the Elbrus-1 multiprocessor nature encourages one to further improve the operation system. Implementations, discussed here (multicomputer and multiprocessor) progressively bring us close to an operation system for multiprocessing. Multicomputer/multiprocessor combination allows to unify a multiprocessor efficiency and multicomputer reliability. The DISPAK OS utilization demonstrated it to be functionally complex and to have a high performance.

V. A. Zherebtsov, M. I. Kaplunov, V. G. Podvalny
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 74-79.

      Experimental data processing has the feature of both business (information capacity) and scientific (complicated problems) applications. Individuals involved in this sphere have various-level programmer skill. Therefore the data processing system proposed contains a wide range of data and processing mode description facilities - from the simplest to extremely complicated ones. The primary system purpose is to formalize and to automate experiment processing applications by sharing operations between experiment makers, mathematicians and computers.

Yu. I. Akutin, N. N. Vyskubenko, A. S. Maksimov, V. I. Samoylov
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 80-83.

      The paper presents a general description of the BSO system implementing centralized output functions within a nonhomogeneous computer complex. The system purpose, main capabilities, functional structures, and operation logic are examined. The background for creationg the system is elucidated. New possibilities, provided by the BSO, are discussed. The system design has taken into acount some disadvantages of earlier local output subsystems. Some new capabilities were also provided including failed output repeatability during several hours after an output process has been completed; detailed system-status information for the operator; simplicity of system function extensions (maintainance of new machine types and new classes of output files). The BSO operation showed that it successfully performs the functions assigned.

A. A. Sadovoy
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1984. No 3. P. 84-93.

      Mathematical basis of multidimensional angular function techniques is analyzed. The paper compares hyperspherical functions, generalized hyperspherical functions, used in atomic nucleus theory, relativistic and nonrelativistic multidimensional angular Coulomb functions for evaluating quantum mechanics system properties with long-range interaction. Sturm representations of Green functions for several multiple particle system were derived to illustrate the efficiency of using mathematical capabilities of multidimensional angular functions. The issues related to using these computation techniques in technical physics are discussed.