A Locally One-Dimensional Scheme for the Distribution Functions Equation by Ice Particles Masses, Considering the Interaction of Droplets and Crystals

Ashabokov, B. A.  , Khibiev, A. Kh.  , Shkhanukov-Lafishev, M. Kh.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 2.

Abstract:
The work is devoted to the construction of a locally one-dimensional difference scheme for calculating the first boundary value problem for a general parabolic equation for the mass distribution function of ice particles. The functions \(u_1(x,z,m,t)\), \(u_2(x,z,m,t)\) are introduced such that \(u_1(x,z,m,t)\,dm\) and \(u_2(x,z,m,t)\,dm\) give at each point \((x,z)\) at time \(t\), the concentration of cloud droplets and ice particles, respectively, whose mass is in the range from \(m\) to \(m+dm.\) The equation is written with respect to the function \(u_2(x,z,m,t) \), the function \(u_1(x,z,m,t) \) (the droplet mass distribution function) is given in the equation. The equation is part of a system of integro-differential equations for the mass distribution functions of droplets and ice particles describing microphysical processes in convective clouds against the background of a given thermohydrodynamics. A locally one-dimensional difference scheme for a general parabolic equation in a \(p\)-dimensional parallelepiped is constructed by the method of total approximation. To describe the interaction of droplets and crystals, nonlocal (nonlinear) integral sources are included in the equation. By the method of energy inequalities, an a priori estimate is obtained, from which the stability and convergence of the difference scheme follow. The results of the work will be used to build a model of microphysical processes in mixed convective clouds, which will be used to conduct research in such topical areas as the study of the role of the system properties of clouds in the formation of their microstructural characteristics and the development of technology for managing precipitation processes in convective clouds by introducing particles of ice-forming reagents.

Keywords: boundary value problem, difference scheme, stability, convergence of the scheme, approximation error.

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