On Model Two-Dimensional Pressureless Gas Flows: Variational Description and Numerical Algorithm Based on Adhesion Dynamics

N. V. Klyushnev; Клюшнев Н. В.; Yu. G. Rykov; Рыков Ю. Г.

doi:10.31857/S0044466923040105

On Model Two-Dimensional Pressureless Gas Flows: Variational Description and Numerical Algorithm Based on Adhesion Dynamics

Authors: Klyushnev N.V.¹, Rykov Y.G.¹
Affiliations:
1. Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
Issue: Vol 63, No 4 (2023)
Pages: 639-656
Section: Mathematical physics
URL: https://ogarev-online.ru/0044-4669/article/view/136168
DOI: https://doi.org/10.31857/S0044466923040105
EDN: https://elibrary.ru/IPHKRD
ID: 136168

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Abstract
About the authors
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Abstract

Weak solutions of the system of pressureless gas dynamics equations in two dimensions are studied. Theoretical issues are considered, namely, the general mathematical theory of conservation laws for the system is addressed. Emphasis is placed on an important distinctive feature of this system: the emergence of strong density singularities along manifolds of different dimensions. This property is characterized as the formation of a hierarchy of singularities. In earlier application-oriented works (e.g., by A.N. Krayko, et al., including more complicated cases of 3D two-phase flows), this property was studied at the physical level of rigor. In this paper, the formation of a hierarchy of singularities is examined mathematically, since, for example, the existence of a solution with a strong singularity at a point (in the 2D case) is rather difficult to prove rigorously. Accordingly, a special numerical algorithm is used to develop mathematical hypotheses concerning solution behavior. Approaches to the construction of a variational principle for weak solutions are considered theoretically. A numerical algorithm based on approximate adhesion dynamics in the multidimensional case is implemented. The algorithm is tested on several examples (2D Riemann problem) in terms of internal convergence and is compared with mathematical results, including those obtained by other authors.

Keywords

conservation laws, pressureless gas dynamics, variation principle, adhesion dynamics, Rankine–Hugoniot relations, hierarchy of singularities

About the authors

N. V. Klyushnev

Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

Email: n_klyushnev@mail.ru
125047, Moscow, Russia

Yu. G. Rykov

Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

Author for correspondence.
Email: n_klyushnev@mail.ru
125047, Moscow, Russia

References

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