and Computational Methods

#### 517.9:532:536 Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions

**Polyanin A. D. (Bauman Moscow State Technical University/Ishlinsky Institute for Problems in Mechanics/MEPhI), Zhurov A. I. (Cardiff University/Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences)**

doi: 10.18698/2309-3684-2015-4-337

We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the formut = [G(u)ux ]x F(u,w),where w = u(x,t) and w = u(x,t ), with denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form =1 = () () N

n n n u x t , withn (x) and n (t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay = (t). We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the formut = div[G(u)u] F(u,w).Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.

Polyanin A., Zhurov A. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 3-37

#### 22.251 Modeling of the process of interaction of the shock wave with cylindrical shell considering wave radiation effect

**Dubrovin V. M. (Bauman Moscow State Technical University), Butina T. A. (Bauman Moscow State Technical University), Polyakova N. S. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-4-3852

The objective of this research is to examine the shock wave with cylindrical shell and to propose a method for calculating its dynamic stability under axial compressive timevarying load. For weak shock waves we conducted comparative analysis of the exact solution and the existing approximate solutions. We evaluated the wave radiation effect after the shell deformation. The case of linearly varying load was considered as an example.

Dubrovin V., Butina T., Polyakova N. Modeling of the process of interaction of the shock wave with cylindrical shell considering wave radiation effect. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 38-52

#### 536.2 Application of the least squares method to the problem of radiation transfer in a spherical cavity

**Zarubin V. S. (Bauman Moscow State Technical University), Pugachev O. V. (Bauman Moscow State Technical University), Savelyeva I. Y. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-4-5365

A lot of heat-shielding materials used in engineering have porous structure. When there is an intensive thermal exposure, there occurs a necessity to consider thermal energy transfer by means of radiation in pores of such materials. We contructed a mathematical model describing heat exchange by radiation in a spherical cavity. Its form can be considered as an average statistical form in relation to forms of closed pores in solid bodies. For the quantitative analysis of this model we used the method of the least squares and introduced an equivalent coefficient of thermal conductivity in the conditional continuous environment filling a pore. This allows to regard the material with porous structure as a continuous non-uniform solid body.

Zarubin V., Pugachev O., Savelyeva I. Application of the least squares method to the problem of radiation transfer in a spherical cavity. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 53-65

#### 629.1.028 The verification of the mathematical model of the multiwheeled vehicles movement with pliable on torsion by bearing system

**Zhileykin M. M. (Bauman Moscow State Technical University), Sarach E. B. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-4-6674

Within solving the problems of active control of elastic and damping elements of multiwheeled vehicles (MWV) suspension, there arises an issue of utmost importance: that of studying the properties of suspension families, designed both for different strokes and different loads. We employed methods of experimental investigation and we conducted the verification of the mathematical model of the multi-wheeled vehicles movement with pliable on torsion by bearing system. We carried out calculation and experimental data analysis which indicates good results.

Zhileykin M., Sarach E. The verification of the mathematical model of the multiwheeled vehicles movement with pliable on torsion by bearing system. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 66-74

#### 519.6:533.6 Application of RKDG method for computational solution of three-dimensional gas-dynamic equations with non-structured grids

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Koryakov M. N. (Bauman Moscow State Technical University), Zakharov A. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-4-7591

This article deals with the finite-element RKDG method (Runge-Kutta Discontinuous Galerkin) and its application for numerical integration of three-dimensional system of equations of ideal gas on unstructured grids. By means of the described algorithm we solved two test tasks. For each task we conducted the analysis and compared the task solution with well-known analytical solutions or with tabular data. We also give error assessment in the solution.

Dimitrienko Y., Koryakov M., Zakharov A. Application of RKDG method for computational solution of three-dimensional gas-dynamic equations with non-structured grids. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 75-91

#### 519.632.4:532.516.5 On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers

**Fomin A. A. (1T. F. Gorbachev Kuzbass State Technical University), Fomina L. N. (Кемеровский государственный университет)**

doi: 10.18698/2309-3684-2015-4-92109

The research explored questions of the convergence of iterative processes and correctness of the solutions on the example of the problem about a steady-state flat square lid-driven cavity flow of incompressible viscous liquid. The problem is solved for Reynolds numbers of 15000 < Re < 20000 and steps of grid 1/128 > h > 1/2048. The findings of the research illustrate that not for all relationships between Re and h the convergence of iterative processes is stable and the resulting steady-state solutions are qualitatively correct. We conducted a qualitative analysis of the solutions of the problem in the coordinate system (Re, 1/h) in terms of the convergence of iterative process, solution correctness and the required computing time. According to the literature and the results of systematic calculations we conclude that the stability of the convergence of iterative process on the coarse grid depends on the degree of influence of the artificial viscosity and/or the condition number of the matrix of difference elliptical linear algebraic equations, and on the detailed grid it depends on the grid Reynolds number. At high Reynolds numbers steady calculations can be carried out either on very coarse grids, or on very detailed ones. The width of the zone of instability in terms of parameter 1/h increases with increasing Reynolds number. Since the coarse grid solution is incorrect, and the use of detailed grid leads to very high costs of computer time, the further increase of the Reynolds number in the problem is associated with increasing the order of approximation of the differential equations.

Fomin A., Fomina L. On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 92-109

#### 551.521 Simulation of atmospheric parameters of two-dimensional fields in problems of laser remote sensing

**Ivanov S. E. (Bauman Moscow State Technical University), Gorodnichev V. A. (Bauman Moscow State Technical University), Belov M. L. (Bauman Moscow State Technical University), Mikhaylovskaya M. B. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2015-4-110121

The objective of this research is to examine the problem of simulating two-dimensional fields of atmospheric backscattering aerosol coefficient and speed of atmospheric wind. The problem is essential for mathematical simulation of laser remote sensing systems and laser location systems. For wind correlation lidar we selected optimal simulation parameters with regard to simulation time and conformity of atmospheric field parameters and their statistical characteristics with set-up statistical characteristics. The findings of the research illustrate that for small sized atmospheric irregularities it is more effective to use the forming filter method, but for large sized atmospheric irregularities the spectral method is the most effective simulation method.

Ivanov S., Gorodnichev V., Belov M., Mikhaylovskaya M. Simulation of atmospheric parameters of two-dimensional fields in problems of laser remote sensing. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 110-121