CONSERVATIVE SCHEMES USING ANTI-DIFFUSIONAL VELOCITIES FOR SOLVING TRANSPORT EQUATION

V. Yu. Gusev, M. Yu. Kozmanov
VANT. Ser.: Mat. Mod. Fiz. Proc 1997. Вып.1. С. 52.

      This work is devoted to constructing monotone difference schemes for the transport equation. The developed methods may be applied, for example, for solving problems of contaminant transport in the atmosphere, radiation transport with account of absorption and scattering.
      The computed results of model problems were presented in comparison with exact solutions.

      One of versions of a new maximum principle base monotonizer is considered. This is used to monotonize solution to difference equations generated by explicit schemes of a high (second and higher) order of accuracy which approximate the equation of transport with a constant coefficient.
      In [1] by monotone difference schemes for the equation of heat conduction were called the schemes satisfying the maximum principle. A similar definition was used in [2,3] for the system of the equation of radiation transport and the equation of energy. This work is further development of the approach from [4] to solution monotonization.
      The essence of the approach proposed consists in the fact that the flows computed at the stage of predictor for the scheme of a high accuracy order secure meeting the maximum principle, namely: u_min ≤ u^j ≤ u_max. Here u_min and u_max are local minimum and maximum of the numerical solution at the time t the vicinity of the point x_j, u_j the numerical solution at the time t_n + τ at the point x_j. When this inequality is violated then the flows are computed from the equation written on the basis of this inequality for the left or right boundary by substitution of u_j with, the expression from the initial difference equation.
      The monotonizer basing on the schemes from [5,6] was used to obtain monotone schemes of the second, third and fourth order of accuracy which are higher than the first-order on the whole and produce numerical solutions of rather a low diffusion as compared to the explicit monotone scheme of the first order of accuracy which is substantiated with the results of numerical solution of various model problems.
      1. Samarsky A.A. Theory of difference schemes. Moscow, Nauka Publishers, 1977.
      2. Andreyev E.S., Kozmanov M. Yu., Rachilov E.B. The maximum principle for the energy equation system and the non-stationary equation of radiation transport // Zhurn. Vych. Mat. i Mat Fiz. 1983. Vol. 23, N 1. P. 151-159.
      3. Kozmanov M.Yu. Monotone schemes for the system of equations of radiation transport // Voprosy Atomnoy Nauki i Tekhniki. Ser. Matematicheskoye Modelirovaniye Fizicheskikh Protsessov. 1989. N 2. P. 51-54.
      4. Gusev V. Yu., Kozmanov M. Yu. Conservative schemes using characteristics and anti-diffusion velocities for solving the equation of transport // Ibid. 199d. N 1-2. P. 24-32.
      5. Moiseev N. Ya. On one modification to the Godunov difference scheme // Voprosy Atomnoy Nauki i Tekhniki. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki. 1983. N 3. P. 35-43.
      6. Moiseyev N.Ya. On one approach to construction of hybrid scheme of an enhanced order of approximation // Ibid.1988. N2. P. 11-17.

      The MIMD multiprocessor MP-X (various instruction threads over various data threads) consisting of 2^x- processing elements is a functional module designed for parallel computations.
      The MP-X rests on the following concepts;
      — combination of the MP-X multiprocessor with the general purpose host computer Y allowing to combine the advantages of parallel, vector and scalar, processing when executing a program that contains non-parallelizable fragments;
      — adaptibility of connectivity structure to the algorithms of the applications to be run (program emulation of various interprocessor connections);
      — design modularity which allows to upgrade the performance according to the user demands a;nd simplifies the transition to new microprocessors and new message passing technologies;
      — hardware support of the routing algorithms to increase the efficiency of the interprocessor transfers;
      — higher failure tolerance level of MP-X due to redundancy;
      — orientation to the common programming languages with the support of parallel program development tools standard in this field.
      The most efficient applications for MP-X-Y include: multidimensional problems in computational physics, design computerization, image processing, artificial intellect and others that require a maximum possible number of operations for this class of Y computers.
      The concepts presented are used in MP-0, MP-3 models and serve the base for MP-7 project.
      The description is given for the component base of existing and future models of some MP-X, as well as the software used for the development of parallel codes. The results are given for some test problems on various types of similar computers including MP-0 and MP-3.
      The report was accompanied by the illustrations arid demonstration of operational models.