# Entropy Correct Spatial Discretizations for 1D Regularized Systems of Equations for Gas Mixture Dynamics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. One-Dimensional (1D) Regularized System of Equations for Dynamics of General Multicomponent Gas Mixtures

**Lemma**

**1.**

**Proof.**

## 3. One-Dimensional (1D) Regularized System of Equations for the Dynamics of One-Velocity Multicomponent Gas Mixtures in the Presence of Diffusion Fluxes

**Proposition**

**1.**

**Proof.**

## 4. A Spatial Discretization of the 1D Regularized System of Equations for the Dynamics of General Gas Mixtures

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 5. A Spatial Discretization of the 1D Regularized System of Equations for the Dynamics of One-Velocity Gas Mixtures in the Presence of Diffusion Fluxes

**Proposition**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 6. Numerical Experiments

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Landau, L.D.; Lifschitz, E.M. Theoretical Physics. Vol. 6. Fluid Mechanics, 2nd ed.; Pergamon Press: Oxford, UK, 1987. [Google Scholar]
- Nigmatulin, R.Y. Dynamics of Multiphase Media; Hemisphere: New York, NY, USA, 1990; Volume 1. [Google Scholar]
- Rajagopal, K.L.; Tao, L. Mechanics of Mixtures; World Scientific: Singapore, 1995. [Google Scholar]
- Giovangigli, V. Multicomponent Flow Modeling; Birkhäuser: Boston, MA, USA, 1999. [Google Scholar] [CrossRef] [Green Version]
- Ruggeri, T.; Sugiyama, M. Classical and Relativistic Rational Extended Thermodynamics of Gases; Springer: Cham, Switzerland, 2021. [Google Scholar] [CrossRef]
- Bisi, M.; Groppi, M.; Martalò, G. Macroscopic equations for inert gas mixtures in different hydrodynamic regimes. J. Phys. A Math. Theor.
**2021**, 54, 085201. [Google Scholar] [CrossRef] - Kulikovskii, A.G.; Pogorelov, N.V.; Semenov, A.Y. Mathematical Aspects of Numerical Solution of Hyperbolic Systems; Chapman and Hall/CRC: New York, NY, USA, 2001. [Google Scholar] [CrossRef] [Green Version]
- LeVeque, R.J. Finite Volume Methods for Hyperbolic Problems; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Abgrall, R.; Shu, C.-W. (Eds.) Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues; Handbook of Numerical Analysis, 17; North Holland: Amsterdam, The Netherlands, 2016. [Google Scholar] [CrossRef]
- Chetverushkin, B.N. Kinetic Schemes and Quasi-Gas Dynamic System of Equations; CIMNE: Barcelona, Spain, 2008. [Google Scholar]
- Elizarova, T.G. Quasi-Gas Dynamic Equations; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar] [CrossRef]
- Sheretov, Y.V. Continuum Dynamics with Spatial-Temporal Averaging; RKhD: Moscow/Izhevsk, Russia, 2009. (In Russian) [Google Scholar]
- Guermond, J.-L.; Popov, B.; Tomov, V. Entropy viscosity method for the single material Euler equations in Lagrangian frame. Comput. Meth. Appl. Mech. Eng.
**2016**, 300, 402–426. [Google Scholar] [CrossRef] [Green Version] - Feireisl, E.; Lukáčová-Medvidová, M.; Mizerová, H. A finite volume scheme for the Euler system inspired by the two velocities approach. Numer. Math.
**2020**, 144, 89–132. [Google Scholar] [CrossRef] [Green Version] - Dolejší, V.; Svärd, M. Numerical study of two models for viscous compressible fluid flows. J. Comput. Phys.
**2021**, 427, 110068. [Google Scholar] [CrossRef] - Elizarova, T.G.; Zlotnik, A.A.; Chetverushkin, B.N. On quasi-gasdynamic and quasi-hydrodynamic equations for binary mixtures of gases. Dokl. Math.
**2014**, 90, 719–723. [Google Scholar] [CrossRef] - Balashov, V.A.; Savenkov, E.B. Quasi-hydrodynamic model of multiphase fluid flows taking into account phase interaction. J. Appl. Mech. Tech. Phys.
**2018**, 59, 434–444. [Google Scholar] [CrossRef] - Kudryashova, T.; Karamzin, Y.; Podryga, V.; Polyakov, S. Two-scale computation of N2–H2 jet flow based on QGD and MMD on heterogeneous multi-core hardware. Adv. Eng. Software.
**2018**, 120, 79–87. [Google Scholar] [CrossRef] - Elizarova, T.G.; Zlotnik, A.A.; Shil’nikov, E.V. Regularized equations for numerical simulation of flows of homogeneous binary mixtures of viscous compressible gases. Comput. Math. Math. Phys.
**2019**, 59, 1832–1847. [Google Scholar] [CrossRef] - Elizarova, T.G.; Shil’nikov, E.V. Numerical simulation of gas mixtures based on the quasi-gasdynamic approach as applied to the interaction of a shock wave with a gas bubble. Comput. Math. Math. Phys.
**2021**, 61, 118–128. [Google Scholar] [CrossRef] - Balashov, V.; Zlotnik, A. On a new spatial discretization for a regularized 3D compressible isothermal Navier–Stokes–Cahn–Hilliard system of equations with boundary conditions. J. Sci. Comput.
**2021**, 86, 33. [Google Scholar] [CrossRef] - Zlotnik, A.; Fedchenko, A. On properties of aggregated regularized systems of equations for a homogeneous multicomponent gas mixture. Math. Meth. Appl. Sci.
**2022**, 45, 8906–8927. [Google Scholar] [CrossRef] - Amosov, A.A.; Zlotnik, A.A. A study of finite-difference method for the one-dimensional viscous heat conductive gas flow equation. Part I: A priori estimates and stability. Sov. J. Numer. Anal. Math. Model.
**1987**, 2, 159–178. [Google Scholar] [CrossRef] - Tadmor, E. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer.
**2003**, 12, 451–512. [Google Scholar] [CrossRef] - Prokopov, G.P. Necessity of entropy control in gasdynamic computations. Comput. Math. Math. Phys.
**2007**, 47, 1528–1537. [Google Scholar] [CrossRef] - Tadmor, E. Entropy stable schemes. In Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues. Handbook of Numerical Analysis; Abgrall, R., Shu, C.-W., Eds.; North Holland: Amsterdam, The Netherlands, 2016; Volume 17, Chapter 18; pp. 467–493. [Google Scholar]
- Carpenter, M.H.; Fisher, T.C.; Nielsen, E.J.; Parsani, M.; Svärd, M.; Yamaleev, N. Entropy stable summation-by-parts formulations for compressible computational fluid dynamics. In Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues. Handbook of Numerical Analysis; Abgrall, R., Shu, C.-W., Eds.; North Holland: Amsterdam, The Netherlands, 2016; Volume 17, Chapter 19; pp. 495–524. [Google Scholar]
- Zlotnik, A.A. Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation. Comput. Math. Math. Phys.
**2012**, 52, 1060–1071. [Google Scholar] [CrossRef] - Gavrilin, V.A.; Zlotnik, A.A. On spatial discretization of the one-dimensional quasi-gasdynamic system of equations with general equations of state and entropy balance. Comput. Math. Math. Phys.
**2015**, 55, 264–281. [Google Scholar] [CrossRef] - Zlotnik, A.A. Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations. Comput. Math. Math. Phys.
**2017**, 57, 706–725. [Google Scholar] [CrossRef] - Zlotnik, A.; Lomonosov, T. Verification of an entropy dissipative QGD-scheme for the 1D gas dynamics equations. Math. Model. Anal.
**2019**, 24, 179–194. [Google Scholar] [CrossRef] [Green Version] - Zhang, C.; Menshov, I.; Wang, L.; Shen, Z. Diffuse interface relaxation model for two-phase compressible flows with diffusion processes. J. Comput. Phys.
**2022**, 466, 111356. [Google Scholar] [CrossRef] - Renac, F. Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations. J. Comput. Phys.
**2021**, 445, 110584. [Google Scholar] [CrossRef] - Abgrall, R.; Karni, S. Computations of compressible multifluids. J. Comput. Phys.
**2001**, 169, 594–623. [Google Scholar] [CrossRef] [Green Version] - Movahed, P.; Johnsen, E. A solution-adaptive method for efficient compressible multifluid simulations, with application to the Richtmyer-Meshkov instability. J. Comput. Phys.
**2013**, 239, 166–186. [Google Scholar] [CrossRef] - Borisov, V.E.; Rykov, Y.G. An exact Riemann solver in the algorithms for multicomponent gas dynamics. Keldysh Inst. Appl. Math. Preprints
**2018**, 96, 1–28. (In Russian) [Google Scholar] [CrossRef] - Bird, G.A. Molecular Gas Dynamics and the Direct Simulation of Gas Flows; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- Ruev, G.A.; Fedorov, A.V.; Fomin, V.M. Description of the anomalous Rayleigh-Taylor instability on the basis of the model of dynamics of a three-velocity three-temperature mixture. J. Appl. Mech. Tech. Phys.
**2009**, 50, 49–57. [Google Scholar] [CrossRef] - Ruev, G.A.; Fedorov, A.V.; Fomin, V.M. Development of the Richtmyer-Meshkov instability upon interaction of a diffusion mixing layer of two gases with shock waves. J. Appl. Mech. Tech. Phys.
**2005**, 46, 307–314. [Google Scholar] [CrossRef] - Boscarino, S.; Cho, S.Y.; Groppi, M.; Russo, G. BGK models for inert mixtures: Comparison and applications. Kin. Relat. Models.
**2021**, 14, 895–928. [Google Scholar] [CrossRef] - Zlotnik, A.; Gavrilin, V. On quasi-gasdynamic system of equations with general equations of state and its application. Math. Model. Anal.
**2011**, 16, 509–526. [Google Scholar] [CrossRef] - Kvasnikov, I.A. Thermodynamics and Statistical Physics. Vol. 1. Theory of Equilibrium Systems: Thermodynamics, 2nd ed.; Editorial URSS: Moscow, Russia, 2002. (In Russian) [Google Scholar]

**Figure 1.**Example 1. The QGD ($a=0.25$, red) and QHD ($a=0.6$, blue) results for $\beta =0.4$, $N=1601$ and $t=0.2$. Hereafter, the blue graphs are almost entirely situated behind the red ones.

**Figure 2.**Example 2. The QGD ($a=0.25$, red) and QHD ($a=0.6$, blue) results for $\beta =0.3$, $N=2001$ and $t=0.2$.

**Figure 3.**Example 3. The QGD ($a=0.25$, red) and QHD ($a=0.6$, blue) results for $\beta =0.1$, $N=4001$ and $t=0.011$.

**Table 1.**The parameters of the initial data to the left and right of the discontinuity between two gases and the final time of computations.

Example | $\mathit{\rho}$ | p | u | $\mathit{\gamma}$ | ${\mathit{t}}_{\mathit{fin}}$ |
---|---|---|---|---|---|

(1) left | 1 | 1 | 0 | 1.4 | 0.2 |

(1) right | 0.125 | 0.1 | 0 | 1.6 | |

(2) left | 1 | 2 | 0 | 2 | 0.2 |

(2) right | 0.125 | 0.1 | 0 | 1.4 | |

(3) left | 1 | 500 | 0 | 1.4 | 0.011 |

(3) right | 1 | 0.2 | 0 | 1.6 |

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Zlotnik, A.; Fedchenko, A.; Lomonosov, T.
Entropy Correct Spatial Discretizations for 1D Regularized Systems of Equations for Gas Mixture Dynamics. *Symmetry* **2022**, *14*, 2171.
https://doi.org/10.3390/sym14102171

**AMA Style**

Zlotnik A, Fedchenko A, Lomonosov T.
Entropy Correct Spatial Discretizations for 1D Regularized Systems of Equations for Gas Mixture Dynamics. *Symmetry*. 2022; 14(10):2171.
https://doi.org/10.3390/sym14102171

**Chicago/Turabian Style**

Zlotnik, Alexander, Anna Fedchenko, and Timofey Lomonosov.
2022. "Entropy Correct Spatial Discretizations for 1D Regularized Systems of Equations for Gas Mixture Dynamics" *Symmetry* 14, no. 10: 2171.
https://doi.org/10.3390/sym14102171