Issue N^{o} 1, 2008 
APPLICATION OF MULTIGRID TECHNIQUES FOR SOLVING PARABOLIC EQUATIONS
M. E. Ladonkina, O. Yu. Milyukova, V. F. Tishkin VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 419.
New effective solution algorithm for parabolic equations based on multigrid technique is suggested which maintains virtues of implicit scheme — stability and accuracy. At the same time this algorithm allows to considerably reduce number of arithmetical operations on each time layer. Absolute stability of the suggested technique is theoretically proved by example of onedimensional model problem. Accuracy of the technique is investigated by example of twodimensional model problem. It is proved that for model problem solution obtained using the suggested technique differs little from solution for implicit scheme on explicit grid. Computations of one and twodimensional problems with continuous and discontinuous coefficients show good accuracy of the suggested technique.

COMPARISON OF DIFFERENT VARIANTS OF HIGHFREQUENCY APPROXIMATION ON THE MODEL PROBLEM FOR MAXWELL’S EQUATIONS A. L Golubev, N. A. Ismailova, V. A. Terekhin VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 2029.
Exact solution of Maxwell’s equations is compared on representative twodimensional model problem with solutions of simplified equations derived from different variants of highfrequency approximation. Comparison shows that for the considered problem derivatives accounting by angle variables in highfrequency approximation equations essentially expands time fields of application of this approximation.

INVESTIGATION OF INFLUENCE OF SOME ARTIFICIAL VISCOSITY FORMS ON “ENTROPY TRACE” CHARACTER FOR NUMERICAL SOLUTION OF GASDYNAMICS EQUATIONS A. M. Stenin, E. A. Solov’eva VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 3041.
The paper considers the problem of errors reduction for values of quantities in entropy trace domain originating in computations of Lagrangian gasdynamics problems conducted with the help of difference schemes of shock waves endtoend computation. Artificial mass diffusion, artificial impulse diffusion and artificial total energy diffusion are conservatively entered into gasdynamics equations set in Euler coordinates. Formulas for velocity of artificial mass diffusion in continuity equations and for artificial viscosities in motion and energy equations in Lagrangian coordinates are derived from this equations set. In the case of cylindrical and spherical symmetry of flows tensorial artificial viscosity similar to physical viscosity is entered to the motion equation instead of scalar viscosity. Methods of modification of obtained viscosity system by means of Hugoniot conditions on shock wave are considered which improve computation results in entropy trace domain. Results of conducted test computations show high efficiency of new viscosity system concerning reduction of entropy trace for numerical solution of Lagrangian gasdynamics equations.

DEBYE D_{3}(x) FUNCTION APPROXIMATION IN 0 ≤ x < ∞ DOMAIN BY CLASS C^{2} FUNCTION D. G. Gordeev, A. L Lomaykin VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 4250.
The paper presents results of fitting and realization of approximation expression for Debye D_{3}(x) function in 0 ≤ x < ∞ domain. Approximation provides continuity up to and including the second derivative and computation time acceptable for application programs.

BILEVEL PARALLELIZATION IN MIXED MEMORY MODEL FOR COMPUTING GASDYNAMICS PROBLEMS IN TIM2D TECHNIQUE A. A. Voropinov, S. S. Sokolov, I. G. Novikov VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 5159.
The paper gives description of parallelization scheme for computing gasdynamics problems subject to elastoplastic material properties using TIM2D technique destined for solving problems of mechanics of continua on nonregular polygonal grids with arbitrary structure. Threelevel parallelization is realized for TIM2D technique. On the first (upper) level computation parallelization over mathematical domains is realized in distributed memory model using messaging interface MPI. On the second level computation inside countable domain is parallelized for paradomains also using MPI (countable domain is splitted onto fragments by singlelayer cell superposition). On the third (lower) level parallelization of counting loop iterations is realized in common memory model using OpenMP interface. These approaches can be used both together in different combinations and separately for computing the same problem. The paper considers in detail parallelization algorithms on the upper and the lower levels. Different modes of parallel computation are described. Measurements of acceleration and efficiency of parallelization are given.

UNIFORM TASK CONTROL SYSTEM FOR HETEROGENEOUS COMPUTER COMPLEX A. B. Kiselev, Yu. G. Bartenev, A. M. Vargin, S. N. Kiselev, S. I. Kolpakov VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 6066.
Architecture and performance capabilities of uniform control system for calculating parallel applications on computers of RFNCVNIIEF are described. System is created in order to organize centralized entry, test and task computation control in heterogeneous computer complex.

ON SOME STEREOTYPES OF PARALLEL PROGRAMMING G. V. Baydin VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 6775.
Consistency of some working ideas of parallel programming is examined. Problems of computations against interchanges, optimal message size, optimality of nonblocking and cooperative operations are considered.

50 TH ANNIVERSARY OF FORTRAN: HISTORY, CURRENT STATE AND FUTURE V. P. Sokolov VANT. Ser.: Mat. Mod. Fiz. Proc. 2008. No 1. P. 7689.
The paper is based on materials of the report represented on mathematical conference of RFNOVNHTF in Snezhinsk in 2007. Fortran can be described both very briefly and in considerable detail so this paper must be considered only as the author’s outline. The author hopes that the paper will attract attention to Fortran and at least reduce distrust and neglect undeservedly originated among programmers in respect to Fortran during the last two decades. Both officially published materials and information from Internet were used while preparing the paper. The author apologizes for possible inaccuracies.

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