Figure 1.
Numerical solution by scheme with limiter (40)
-
Previous Article
Pricing down-and-out power options with exponentially curved barrier
- NACO Home
- This Issue
-
Next Article
Preface
September
2018, 8(3): 277-289.
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 |
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.
Mathematics Subject Classification:
Primary: 65N12.
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)
: 277-289.
doi: 10.3934/naco.2018017
![]()
References:
| [1] |
R. D. Aloev, Z. K. Eshkuvatov, Sh. 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. Aloev, A. M. Blokhin and M. U. Hudayberganov, One class of stable difference schemes for hyperbolic system, American Journal of Numerical Analysis, 2 (2014), 85-89. Google Scholar |
| [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. Aloev, Z. K. Eshkuvatov, Sh. 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. Aloev, A. M. Blokhin and M. U. Hudayberganov, One class of stable difference schemes for hyperbolic system, American Journal of Numerical Analysis, 2 (2014), 85-89. Google Scholar |
| [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 2.
Line is exact solution, point-numerical solution by scheme (40)
| [1] |
Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286 |
| [2] |
Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495 |
| [3] |
Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035 |
| [4] |
Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34 |
| [5] |
Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015 |
| [6] |
Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045 |
| [7] |
Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864 |
| [8] |
Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385 |
| [9] |
Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221 |
| [10] |
Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151 |
| [11] |
Raimund Bürger, Kenneth H. Karlsen, John D. Towers. On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 461-485. doi: 10.3934/nhm.2010.5.461 |
| [12] |
Allaberen Ashyralyev. Well-posedness of the modified Crank-Nicholson difference schemes in Bochner spaces. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 29-51. doi: 10.3934/dcdsb.2007.7.29 |
| [13] |
Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks & Heterogeneous Media, 2008, 3 (1) : 1-41. doi: 10.3934/nhm.2008.3.1 |
| [14] |
Lih-Ing W. Roeger. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 415-429. doi: 10.3934/dcdsb.2008.9.415 |
| [15] |
Irena Pawłow, Wojciech M. Zajączkowski. Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 441-466. doi: 10.3934/dcdss.2011.4.441 |
| [16] |
Sijia Zhong, Daoyuan Fang. $L^2$-concentration phenomenon for Zakharov system below energy norm II. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1117-1132. doi: 10.3934/cpaa.2009.8.1117 |
| [17] |
Lars Grüne, Vryan Gil Palma. Robustness of performance and stability for multistep and updated multistep MPC schemes. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4385-4414. doi: 10.3934/dcds.2015.35.4385 |
| [18] |
Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501 |
| [19] |
Jiaxiang Cai, Juan Chen, Bin Yang. Fully decoupled schemes for the coupled Schrödinger-KdV system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5523-5538. doi: 10.3934/dcdsb.2019069 |
| [20] |
Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]