# Finite difference method for numerical solution of the initial and boundary value problem for boltzmann’s sixmoment system of equations

### Abstract

Boltzmann’s one-dimensional non-linear non-stationary moment system of equations in the third approximation is presented, in which the first, third and fourth equations corresponds to the laws of conservation of mass, momentum and energy, respectively. This system contains six equations and represents a nonlinear system of hyperbolic type equations. For the Boltzmann’s six- moment system of equations an initial and boundary value problem is formulated. The macroscopic boundary condition contains the moments of the incident particles distribution function on the boundary and moments of the reflected particles distribution function from the boundary. The boundary condition depends on the temperature of the wall (boundary).

In this work, using the finite-difference method, an approximate solution of the mixed problem for the Boltzmann system of moment equations is constructed in the third approximation under the boundary conditions obtained by approximating the Maxwell boundary condition. For given values of the coefficients included in the moments of the nonlinear collision integral and the parameter depending on the wall temperature, as well as for fixed values of the initial conditions, a numerical experiment was carried out. As a result, the approximate values of the particle distribution function incident on the boundary and reflected from the boundary, as well as the density, temperature and average velocity of gas particles, as moments of the particle distribution function, are obtained.

### References

[1] Grad G., "Kinetic theory of rarefied gases" , Comm. Pure Appl. Math 2 (331) (1949).

[2] Grad G., "Principle of the kinetic theory of gases" , Handuch der Physik, Springer, Berlin 12 (1958): 205–294.

[3] Sakabekov A., "Initial-boundary value problems for the Boltzmann’s moment system equations in an arbitrary approximation" , Sb. Russ. Acad. Sci. Math. 77(1) (1994): 57–76.

[4] Sakabekov A., Initial-boundary value problems for the Boltzmann’s moment system equations (Gylym, Almaty, 2002).

[5] Cercignani C., Theory and application of Boltzmann’s equation (Milano, Italy, 1975).

[6] Kogan M.N., Dynamic of rarefied gas (Moscow, Nauka, 1967): 440.

[7] Kumar K., "Polynomial expansions in Kinetic theory of gases" , Annals of physics 57 (1966): 115–141.

[8] Barantcev R.G., Interaction of rarefied gases with streamlined surfaces (Мoscow, Nauka, 1975): 343.

[9] Sakabekov A., Auzhani Y., "Boundary conditions for the onedimensional nonlinear nonstationary Boltzmann’s moment system equations" , J. Math. Phys. 55 (2014): 123507.

[10] Sakabekov A., Auzhani Y., "Boltzmann’s Six-Moment One-Dimensional Nonlinear System Equations with the Maxwell- Auzhan Boundary Conditions", Hindawi Publishing Corporation Journal of Applied Mathematics 5834620 (2016): 8. http://dx.doi.org/10.1155/2016/5834620.

[11] Auzhani Y., Sakabekov A., "Mixed value problem for nonstationary nonlinear one-dimensional Boltzmann moment system of equations in the first and third approximations with macroscopic boundary conditions" , Kazakh Math. Journal 20(1) (2020): 54–66.

[12] Samarskiy A.A., Difference sheme theory (Мoscow, Nauka 1977): 656.

[13] Richtmyer Robert D., Morton K.W., Difference methods for boundary value problems (Мoscow, 1972): 418.

[2] Grad G., "Principle of the kinetic theory of gases" , Handuch der Physik, Springer, Berlin 12 (1958): 205–294.

[3] Sakabekov A., "Initial-boundary value problems for the Boltzmann’s moment system equations in an arbitrary approximation" , Sb. Russ. Acad. Sci. Math. 77(1) (1994): 57–76.

[4] Sakabekov A., Initial-boundary value problems for the Boltzmann’s moment system equations (Gylym, Almaty, 2002).

[5] Cercignani C., Theory and application of Boltzmann’s equation (Milano, Italy, 1975).

[6] Kogan M.N., Dynamic of rarefied gas (Moscow, Nauka, 1967): 440.

[7] Kumar K., "Polynomial expansions in Kinetic theory of gases" , Annals of physics 57 (1966): 115–141.

[8] Barantcev R.G., Interaction of rarefied gases with streamlined surfaces (Мoscow, Nauka, 1975): 343.

[9] Sakabekov A., Auzhani Y., "Boundary conditions for the onedimensional nonlinear nonstationary Boltzmann’s moment system equations" , J. Math. Phys. 55 (2014): 123507.

[10] Sakabekov A., Auzhani Y., "Boltzmann’s Six-Moment One-Dimensional Nonlinear System Equations with the Maxwell- Auzhan Boundary Conditions", Hindawi Publishing Corporation Journal of Applied Mathematics 5834620 (2016): 8. http://dx.doi.org/10.1155/2016/5834620.

[11] Auzhani Y., Sakabekov A., "Mixed value problem for nonstationary nonlinear one-dimensional Boltzmann moment system of equations in the first and third approximations with macroscopic boundary conditions" , Kazakh Math. Journal 20(1) (2020): 54–66.

[12] Samarskiy A.A., Difference sheme theory (Мoscow, Nauka 1977): 656.

[13] Richtmyer Robert D., Morton K.W., Difference methods for boundary value problems (Мoscow, 1972): 418.

How to Cite

AKIMZHANOVA, Sh.; YESBOTAYEVA, G.; SAKABEKOV, A..
Finite difference method for numerical solution of the initial and boundary value problem for boltzmann’s sixmoment system of equations.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 116, n. 4, dec. 2022. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1153>. Date accessed: 28 jan. 2023. doi: https://doi.org/10.26577/JMMCS.2022.v116.i4.07.
Section

Applied Mathematics

Keywords
Boltzmann’s moment system of equations, microscopic Maxwell boundary condition, macroscopic Maxwell-Auzhan boundary conditions