scholarly journals On L2-dissipativity of a linearized difference scheme on staggered meshes with a quasi-hydrodynamic regularization for 1D barotropic gas dynamics equations

2021 ◽  
pp. 1-27
Author(s):  
Alexander Anatolievich Zlotnik ◽  
Timofey Alexandrovich Lomonosov
Keyword(s):  
Difference Scheme ◽  
Gas Dynamics ◽  
Sufficient Conditions ◽  
Viscosity Coefficient ◽  
Neumann Condition ◽  
Gas Dynamics Equations ◽  
Von Neumann ◽  
Barotropic Gas ◽  
Constant Solution ◽  
The Cauchy Problem

We study an explicit two-level finite difference scheme on staggered meshes, with a quasi-hydrodynamic regularization, for 1D barotropic gas dynamics equations. We derive necessary conditions and sufficient conditions close to each other for L<sup>2</sup>-dissipativity of solutions to the Cauchy problem for its linearization on a constant solution, for any background Mach number M. We apply the spectral approach and analyze matrix inequalities containing symbols of symmetric matrices of convective and regularizing terms. We consider the cases where either the artificial viscosity coefficient or the physical viscosity one is used. A comparison with the spectral von Neumann condition is also given for M=0.

On Some Finite Difference Scheme for Gas Dynamics Equations

2018 ◽  
Vol 73 (4) ◽  
pp. 143-149
Author(s):  
A. V. Zvyagin ◽  
G. M. Kobelkov ◽  
M. A. Lozhnikov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Gas Dynamics ◽  
Gas Dynamics Equations

scholarly journals On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2184
Author(s):  
Alexander Zlotnik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Gas Dynamics ◽  
System Of Equations ◽  
Time Step ◽  
Spatial Direction ◽  
Barotropic Gas ◽  
Rectangular Mesh ◽  
2D And 3D

We deal with 2D and 3D barotropic gas dynamics system of equations with two viscous regularizations: so-called quasi-gas dynamics (QGD) and quasi-hydrodynamics (QHD) ones. The system is linearized on a constant solution with any velocity, and an explicit two-level in time and symmetric three-point in each spatial direction finite-difference scheme on the uniform rectangular mesh is considered for the linearized system. We study L2-dissipativity of solutions to the Cauchy problem for this scheme by the spectral method and present a criterion in the form of a matrix inequality containing symbols of symmetric matrices of convective and regularizing terms. Analyzing these inequality and matrices, we also derive explicit sufficient conditions and necessary conditions in the Courant-type form which are rather close to each other. For the QHD regularization, such conditions are derived for the first time in 2D and 3D cases, whereas, for the QGD regularization, they improve those that have recently been obtained. Explicit formulas for a scheme parameter that guarantee taking the maximal time step are given for these conditions. An important moment is a new choice of an “average” spatial mesh step ensuring the independence of the conditions from the ratios of the spatial mesh steps and, for the QGD regularization, from the Mach number as well.

scholarly journals VERIFICATION OF AN ENTROPY DISSIPATIVE QGD-SCHEME

2019 ◽  
Vol 24 (2) ◽  
pp. 179-194 ◽  
Author(s):  
Alexander Zlotnik ◽  
Timofey Lomonosov
Keyword(s):  
Gas Dynamics ◽  
Convergence Rates ◽  
Necessary Conditions ◽  
Practical Relevance ◽  
L1 Norm ◽  
Gas Dynamics Equations ◽  
Nonlinear Statement ◽  
The Cauchy Problem ◽  
Practical Convergence ◽  
Explicit Finite Difference

An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the previously constructed QGD-schemes and its merits are noticed. Practical convergence rates in the mesh L1-norm are computed. We also analyze the practical relevance in the nonlinear statement as the Mach number grows of recently derived necessary conditions for L2-dissipativity of the Cauchy problem for a linearized QGD-scheme.

Generalized Riemann waves and their adjoinment through a shock wave

2018 ◽  
Vol 13 (2) ◽  
pp. 22 ◽  
Author(s):  
A. Chaiyasena ◽  
W. Worapitpong ◽  
S.V. Meleshko
Keyword(s):  
Shock Wave ◽  
Gas Dynamics ◽  
Lagrange Equation ◽  
Sufficient Conditions ◽  
Lagrangian Description ◽  
Gas Dynamics Equations ◽  
Simple Waves ◽  
Eulerian Descriptions ◽  
Necessary And Sufficient ◽  
Riemann Waves

Generalized simple waves of the gas dynamics equations in Lagrangian and Eulerian descriptions are studied in the paper. As in the collision of a shock wave and a rarefaction wave, a flow becomes nonisentropic. Generalized simple waves are applied to describe such flows. The first part of the paper deals with constructing a solution describing their adjoinment through a shock wave in Eulerian coordinates. Even though the Eulerian form of the gas dynamics equations is most frequently used in applications, there are advantages for some problems concerning the gas dynamics equations in Lagrangian coordinates, for example, of being able to be reduced to an Euler–Lagrange equation. Through the technique of differential constraints, necessary and sufficient conditions for the existence of generalized simple waves in the Lagrangian description are provided in the second part of the paper.

Systematization of relaxation gas dynamics equations. A review

2010 ◽  
Vol 45 (4) ◽  
pp. 517-536
Author(s):  
V. S. Galkin ◽  
S. A. Losev
Keyword(s):  
Gas Dynamics ◽  
Gas Dynamics Equations
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