## Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

#### Article Title

### Using an a priori estimate for constructing difference schemes for quasi-linear hyperbolic systems

#### Abstract

In this paper we consider a class of quasi-linear hyperbolic systems, which allows the construction of a dissipative energy integrals. In the basis of the design and investigation the stability of difference schemes for the numerical solution of the initial boundary value problems for the above class of quasi-linear hyperbolic systems, we put the existence of a discrete analogue of the dissipative energy integrals.

#### First Page

1

#### Last Page

13

#### References

1. Blokhin A. M., Aloev R. D. Energy integrals and their applications to investigation of stability of difference schemes. Novosibirsk, 1993. – 224 p.

2. Godunov S. K. Equations of mathematical physics. Nauka, Moscow, 1979 (in Russian). – 392 p.

3. Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L. Methods of spline functions. Moscow, Nauka, 1980 (in Russian). – 352 p.

4. Godunov S. K. An interesting class of quasi-linear systems. Dokl. Akad. Nauk SSSR, 1961, V.139, No.3, pp.521–523 (in Russian).

5. Godunov S. K., Romenskii E. I. Elements of continuum mechanics and conservation laws. Springer-Verlag, New York, 2003. – 258 p.

6. Vol'pert A. I., Khudyaev S. I. On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sb., 1972, V.16, No.4, pp.517–544.

7. Harten A. On the symmetric form of systems of conservation laws with entropy. J. Comp. Phys., 1983, V.49, pp.151–164.

8. Aloev R. D., Eshkuvatov Z. K., Davlatov Sh. O., Nik Long N. M. A. Sufficient condition of stability of finite element method for symmetric t-hyperbolic systems with constant coefficients. Computers and Mathematics with Applications, 2014, V.68, No.10, pp.1194–1204.

9. Aloev R. D., Blokhin A. M., Hudayberganov M. U. One Class of Stable Difference Schemes for Hyperbolic System, American Journal of Numerical Analysis, 2014, V.2, No.3, pp.85–9.

#### Recommended Citation

Aloev, Rakhmatillo and Khudayberganov, Mirzoali
(2018)
"Using an a priori estimate for constructing difference schemes for quasi-linear hyperbolic systems,"
*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*: Vol. 1:
Iss.
1, Article 7.

DOI: https://doi.org/10.56017/2181-1318.1006