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- General activity
- Subdepartment of applied hydrophysic
- Subdepartment of analytical & numerical methods of mathematical physics (the text is under construction)
- Main recent publications

- Investigations in the areas of numerical analysis, the Monte-Carlo methods, the boundary value problems theory, the theory of functions of complex variables and functional spaces aimed at solving problems of applied mathematical physics.
- Development of high accuracy numerical & analytical methods for equations of mathematical physics including equations of fluid dynamics, magnetohydrodynamics, elasticity, electrodynamics etc.
- Solving applied problems concerning numerical simulation complicated physical and mechanical phenomena.

- Head Prof. A.I.Tolstykh
- Senior researcher Prof. Yanitskii

The main section activity concerns with development, justification and applications of high-order difference schemes based on compact non-centered (upwind) differencing (CUD) for hyperbolic parts of convection-diffusion equations, special emphasize being placed on incompressible and compressible Navier-Stokes and Reynolds equations.

The computational practice and comparisons with traditional schemes have shown that such an approach can provide increase of accuracy for fixed grid point numbers measured by order (or orders) of magnitude. As a rule, no monotonization devices are needed with applying CUD methods.

The recent ideas and their implementations include:

- Construction of arbitrary-order compact upwind approximations via linear combinations of "elementary" CUD operators of fixed order (say, third-order). Such "multioperator" schemes can be efficiently realized by using parallel processing without additional computational expenses.
- Development of high-order time integrators for unsteady problems and marching algorithms.
- Development of combined methods with CUD and radial basis function approach allowing to obtain high-accuracy solutions for complicated geometries (including irregular iced surfaces) using domain decomposition approach.
- Development of new types of "weighted" Monte-Carlo techniques for investigation of gas flows fluctuations.

The developed techniques are applied (or planned to be applied) to CFD problems.

They are, in particular:

- long-range vortex propagation with possible loss of their stability
- DMS and LES for turbulent flows
- shock-wave propagation in turbulent flows.

Applied problems which the division concerns with are:

- Numerical simulation of 2D and 3D incompressible and compressible flows described by the Navier-Stokes or the Reynolds - averaged Navier-Stokes equations (in particular, transonic turbine flows).
- Problems of physical aerodynamics (that is, problems requiring physics + numerics). They include:
- Numerical simulation of aerooptics and aero acoustic turbulent fields near open parts of airborne observatories
- Problems of multiphase flows with droplets causing icing of aircraft elements with subsequent changes of their effective surfaces and aerodynamic characteristics. The investigations are based on advanced physical models (developing in collaboration with the high-level specialist in the aerophysics area) and specially designed numerical techniques.
- Investigations of amplification of turbulent pulsations due to shock waves and shock waves attenuation.

- A.I. Tolstykh, High Accuracy Non-centered Compact Difference Schemes for Fluid Dynamics Applications. World Scientific, p. 314, (1994).
- A.I. Tolstykh and D.A. Shirobokiv, Fifth order compact upwind method and its applications to three-dimensional compressible Navier-Stokes equations, Computational Fluid Dynamics Journal, N 5, pp. 425-438, (1997).
- A.I. Tolstykh, M.V. Lipavskii, On performance of methods with third- and fifth-order compact upwind differencing. J. Comput. Phys. N 140, pp. 205-232 (1998).
- A.D.Savel'ev, Implicit method for turbulent compressible flow c calculations, N 38 c. 522-533 (1998).
- V.N.Koterov, A.D.Savel'ev, A.I.Tolstykh’ Numerical simulation of aerooptics fields near open ports of airborne observatory (in Russian), Matematicheskoe modelirovanie, N 9,p. 27-39 (1997).
- A.I.Tolstykh, Two-Step fifth-order methods for evolutionary problems with positive operators. Positivity N 2, pp. 193-219 (1998).
- Yanitskii V.E., Serikov V.V. Multicomponent rarefied gas weighting algorithm for Monte Carlo simulation. In Proc. 2
^{nd}Japan-Soviet Union Joint CFD Symposium. Tsukuba, Japan. 1990. V. 2 P. 36-43. - Yanitskii V.E. Operator approach to direct Monte Carlo simulation theory in rarefied gas dynamics. In Proc. 17
^{th}Int. RGD Symposium, Aachen,Germany, 1991, P. 770-777. - Yanitskii V.E. Stochastic Model of a Boltzmann Gas and Its Numerical realization. In book: Modern Problems in Computational Aerohydrodynamics. Ed. A.A.Dorodnicyn and P.I.Chushkin. Mir Publishers, Moscow, 1992. P. 339-355.
- Yanitskii V.E., Ivanov S.A. Direct Monte Carlo simulation of free turbulence. In Proc. 5
^{th}Japan-Soviet Union Joint CFD Symposium. Sendai, Japan, 1993. V. 2. P. 385-390. - Azarova O.A., Samsonov A.V., Shtemenko L.S., Shugaev F.V. and Yanitskii V.E. Motion of a shock wave through a gas with random inhomogeneities. Proc. of the Symp. on Shock waves. (March, 14-16, 1996. Tokyo, Japan) Tokyo University of Technology, 1996. P. 579-581.
- Azarova O.A. and Yanitskii V.E. Shock wave propagation in gases with random non-uniformities. Proc. of the 20
^{th}Int. RGD Symp. Beijing, China, Peking University Press, 1996.

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