| Our group develops methods for solving applied problems based on mathematical models which footing is non-polarized radiation or neutron transport equation. For the majority of applied problems the transport equation should be coupled with medium describing equations. For radiation transport these equations are fluid dynamics equations. For neutron transport they are reactor kinetics, fuel burn-up and control equations. Linear transport equation is one of the simplest partial differential equation and its mathematical theory is well known. But its numerical solving still sufficiently difficult due to large number of dimensions and slow convergence of scattering and fission sources iterations. The major work of our group could be described as efficient transport equation dimensionality reduction. On the whole: - Distribution function, which should be found as a solution of a transport equation, depends on time, three spatial coordinates, two angular coordinates, and energy. There are no derivatives with respect to angular and energy variables, but consideration of a distribution function dependence on them is quite memory demanding. Suggested by Goldin in 1964 «quasidiffusion» method enables to convolute angular dependence. Efficient algorithms for solving transport equation coupled with medium characterizing equations could be constructed by use of auxiliary low-order transport equations.
- Particle absorption coefficient dependence on energy is complex with multiple (millions) resonance lines, i.e. in energy-close points absorption could vary by several orders. If every line resolved by ten energy points, in line-by-line type methods computations one gets tens of millions points of discretization in energy variable. There are several approaches to costs reduction of distribution function representation in energy variable: radiance transfer, multigroup method, separation of variables. One of the abovementioned approaches based on Lebesgue averaging over spectrum is developed in our group under supervising of Alexander Shilkov.
- Linear transport equation is an integro-differential one, with the scattering integral in the right-hand side. The scattering indicatrix could have singularities in angles. In case of such strong anisotropy a lot of expansion in Legendre polynomial series terms are required, moreover that series could have sufficiently low convergence rate. Method of strong anisotropy accounting was suggested by Vladimir Goldin and Elena Aristova. In addition the quasidiffusion method considerably accelerates iterations convergence.
- Approximation of transport differential operator in three-dimensional space is designed on high-order and non-oscillation. Approaches used for differential operator approximations construction are: long characteristics, short characteristics, finite elements methods, and so forth.
- Development and realization of efficient high-power algorithms for solving transport equation methods is one of our group goals.
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