September 2018, 8(3): 287-299. doi: 10.3934/naco.2018017

Construction and research of adequate computational models for quasilinear hyperbolic systems

1. 

National University of Uzbekistan, Faculty of Mathematics, 4, Universitetskaya str., Tashkent, Uzbekistan

2. 

Sobolev Institute of Mathematics, 4, Acad. Koptyug str., Novosibirsk, Russia

* Corresponding author: Aloev Rakhmatillo, aloev@mail.ru

Received  April 2017 Revised  October 2017 Published  June 2018

In the paper, we study a class of three-dimensional quasilinear hyperbolic systems. For such system, we set the initial boundary value problem and construct the energy integral. We construct the difference scheme and obtain an a priori estimate for its solution.

Citation: Aloev Rakhmatillo, Khudoyberganov Mirzoali, Blokhin Alexander. Construction and research of adequate computational models for quasilinear hyperbolic systems. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 287-299. doi: 10.3934/naco.2018017
References:
[1]

R. D. AloevZ. K. EshkuvatovSh. O. Davlatov and N. M. A. Nik Long, Sufficient condition of stability of finite element method for symmetric t-hyperbolic systems with constant coefficients, Computers and Mathematics with Applications, 68 (2014), 1194-1204. doi: 10.1016/j.camwa.2014.08.019.

[2]

R. D. AloevA. M. Blokhin and M. U. Hudayberganov, One class of stable difference schemes for hyperbolic system, American Journal of Numerical Analysis, 2 (2014), 85-89.

[3]

A. M. Blokhin and R. D. Aloev, Energy Integrals and Their Applications to Investigation of Stability of Difference Schemes, Novosibirsk, 1993. 224 p(in Russian).

[4]

S. K. Godunov, Equations of Mathematical Physics, Nauka, Moscow, 1979 (in Russian).

[5]

S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521–523. (in Russian).

[6]

S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws, Springer-Verlag, New York, 2003. doi: 10.1007/978-1-4757-5117-8.

[7]

A. Harten, On the symmetric form of systems of conservation laws with entropy, J. Comp. Phys., 49 (1983), 151-164. doi: 10.1016/0021-9991(83)90118-3.

[8]

A. I. Vol'pert and S. I. Khudyaev, On the Cauchy problem for composite systems of nonlinear differential equations, Math. USSR Sb., 16 (1972), 517-544. doi: 10.1070/SM1972v016n04ABEH001438.

[9]

Yu. S. Zavyalov, B. I. Kvasov and V. L. Miroshnichenko, Methods of Spline Functions, Moscow, Nauka, 1980 (in Russian).

show all references

References:
[1]

R. D. AloevZ. K. EshkuvatovSh. O. Davlatov and N. M. A. Nik Long, Sufficient condition of stability of finite element method for symmetric t-hyperbolic systems with constant coefficients, Computers and Mathematics with Applications, 68 (2014), 1194-1204. doi: 10.1016/j.camwa.2014.08.019.

[2]

R. D. AloevA. M. Blokhin and M. U. Hudayberganov, One class of stable difference schemes for hyperbolic system, American Journal of Numerical Analysis, 2 (2014), 85-89.

[3]

A. M. Blokhin and R. D. Aloev, Energy Integrals and Their Applications to Investigation of Stability of Difference Schemes, Novosibirsk, 1993. 224 p(in Russian).

[4]

S. K. Godunov, Equations of Mathematical Physics, Nauka, Moscow, 1979 (in Russian).

[5]

S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521–523. (in Russian).

[6]

S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws, Springer-Verlag, New York, 2003. doi: 10.1007/978-1-4757-5117-8.

[7]

A. Harten, On the symmetric form of systems of conservation laws with entropy, J. Comp. Phys., 49 (1983), 151-164. doi: 10.1016/0021-9991(83)90118-3.

[8]

A. I. Vol'pert and S. I. Khudyaev, On the Cauchy problem for composite systems of nonlinear differential equations, Math. USSR Sb., 16 (1972), 517-544. doi: 10.1070/SM1972v016n04ABEH001438.

[9]

Yu. S. Zavyalov, B. I. Kvasov and V. L. Miroshnichenko, Methods of Spline Functions, Moscow, Nauka, 1980 (in Russian).

Figure 1.  Numerical solution by scheme with limiter (40)
Figure 2.  Line is exact solution, point-numerical solution by scheme (40)
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