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113. Shishmarev K. A., Khabakhpasheva T. I., Korobkin A. The response of ice cover to aload moving along a frozen
channel // Applied Ocean Research. 2016. V.59. pp. 313-326.
Solution of advection-diffusion-reaction problems on a sphere
Yu. N. Skiba
Universidad Nacional Autonoma de Mexico
Email: skiba@unam.mx
DOI 10.24412/cl-35065-2021-1-01-51
The new algorithm proposed in [1] is applied for solving linear advection-diffusion-reaction problems and
nonlinear diffusion problems on a sphere. Discretization of differential problems in space is performed by the
finite volume method using the Gauss theorem for each grid cell. For time discretization, the method of sym-
metrized two-cycle componentwise splitting and the Crank-Nicholson scheme are used. The obtained numeri-
cal method has the second order of approximation in space and time. It is implicit and unconditionally stable;
in addition, operator splitting provides a direct (non-iterative) and fast implementation of all implicit schemes.
The theoretical results are confirmed numerically by simulating various linear processes on the sphere (diffu-
sion in a spherical sector, diffusion flux through the pole, advection flux through the pole, dispersion of a pol-
lutant emitted from multiple point sources) and nonlinear diffusion processes (spiral waves, nonlinear tem-
perature waves, HS, LS and S blow-up combustion modes and solutions of the Gray-Scott model). Numerical
experiments [2] show a high accuracy and efficiency of the method that correctly describes the processes of
advection-diffusion on a sphere (including processes near the poles and through the poles) and the mass bal-
ance of matter in forced and dissipative discrete systems. Moreover, in the absence of external forcing and
dissipation, the method conserves both the total mass and the L2-norm of the solution.
The author thanks the National System of Researchers (SNI, CONACYT, Mexico) for grant 14539.
References
1. Skiba Yu. N. A non-iterative implicit algorithm for the solution of advection�diffusion equation on a sphere // Int. J.
Numer. Methods Fluids. 2015. V. 78, N. 5. P. 257-282.
2. Skiba Yu. N., Cruz-Rodriguez R.C., Filatov D.M. Solution of linear and nonlinear advection-diffusion problems on a
sphere // Numer Methods Partial Differential Eq. 2020. V. 36, N. 6. P. 1922-1937.
Mathematical modeling of the flow in a wake behind an impact of an elastic body on a thin liquid layer
K. A. Shishmarev1, K. E. Naydenova1, T. I. Khabakhpasheva2, A. A. Korobkin3
1Altai State University, Barnaul
2Lavrentyev Institute of Hydrodynamics SB RAS
3University of East Anglia, Norwich, UK
Email: shishmarev.k@mail.ru
DOI 10.24412/cl-35065-2021-1-01-52
The study is motivated by the results of experiments on droplet deposition in an annular gas-liquid flow
and mass transfer between gas and liquid film [1]. A two-dimensional problem of the impact of an elastic body
on a thin layer of liquid is considered. The body is modeled by a thin plate. At the beginning of impact the liq-
uid layer is at rest. At the initial moment of time, the elastic plate touches the liquid at a single point. After the
impact, the plate moves in the positive x-direction and penetrates into the liquid layer (Oxy is the Cartesian
coordinate system). The coupled problem of deflections of an elastic plate and the fluid motion under the
plate is studied in [2] in the leading approximation. After determining the deflections of the plate and the be-
havior of the fluid under the plate, the problem of fluid dynamics in the wake is solved by the method of char-
acteristics and asymptotic methods. Similar problems are studied in [3]. The results of the numerical and ana-
lytical solution of the formulated problem are presented. The main focus of the report is given to the first
terms of the asymptotic decomposition and the jet formations on the free surface in the wake.
The work is supported by Russian Science Foundation, project 19-19-00287.
References
1. Cherdantsev A.V., Hann D.B., Hewakandamby B.N., Azzopardi B.J. Study of the impacts of droplets deposited from
the gas core onto a gas-sheared liquid film // International J. of Multiphase Flow. 2017. V.88. pp.69�86.
2. Khabakhpasheva T.I., Korobkin A.A. Oblique elastic plate impact on thin liquid layer // Physics of Fluids. 2020.
V. 32. 062101.
3. Edwards C.M., Howison S.D., Ockendon H., Ockendon J.R. Non-classical shallow water flows // IMA J. of Applied
Mathematics. 2008. V.73 pp.137�157.
A numerical modelling of urban air quality with mesoscale atmospheric models
A. V. Starchenko1,2, E. A. Shelmina1, L. I. Kizhner1, E. A. Danilkin1,2, E. A. Strebkova2
1Tomsk State University
2V. E. Zuev Institute of Atmospheric Optics SB RAS
Email: starch@math.tsu.ru
DOI 10.24412/cl-35065-2021-1-01-54
The results of the numerical prediction of meteorological parameters and some indicators of atmospheric
air quality in the city of Tomsk using the mesoscale numerical weather forecast model TSUNM3 [1] and the
model of pollutant transport taking into account chemical reactions [2] are presented. For individual historical
dates, the study of the relationship between meteorological conditions and air quality in the city was carried
out using mathematical modeling methods. Particular attention is paid to the formation of aerosol particles in
the atmosphere of the city. The calculation results are compared with the observational data obtained at the
TOR-station of the IAO SB RAS, and with the calculations of the CAMx air quality model.
The work was supported by the Russian Science Foundation (project no. 19-71-20042).
References
1. Starchenko A.V., Bart A.A., Kizhner L.I., Danilkin E.A. Mesoscale meteorological model tsunm3 for the study and
forecast of meteorological parameters of the atmospheric surface layer over a major population center // Tomsk State
University J. of Mathematics and Mechanics. 2020. V. 66. P. 35�55.
2. Starchenko A., Shelmina E., Kizhner L. Numerical Simulation of Meteorological Conditions and Air Quality above
Tomsk, West Siberia // Atmosphere. 2020. V. 11, N. 11. P. 1-15.
