Abstract
A previously formulated new approach to the consideration of systems of quasilinear hyperbolic equations on the basis of variational principles is described in more detail in the case of special systems of three equations. It is shown that each field of characteristics can be represented as a solution of a variational problem. Moreover, the Rankine–Hugoniot relations at the corner points of the characteristics or at the intersections of the characteristics of a single family hold automatically. In the simplest case of the Hopf equation, a numerical algorithm is constructed on the basis of a variational principle.
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References
A. I. Vol’pert, “The spaces BV and quasilinear equations,” Math. USSR Sb. 2 (2), 225–267 (1967).
S. N. Kruzhkov, “First order quasilinear equations in several independent variables,” Math. USSR Sb. 10 (2), 217–243 (1970).
B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978; Am. Math. Soc., Providence, 1983).
J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer Science, New York, 1994).
S. K. Godunov, Numerical Solution of Multidimensional Gas Dynamics Problems (Nauka, Moscow, 1976) [in Russian].
A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).
Yu. G. Rykov, Preprint No. 62, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2011); http://keldysh.ru/papers/2011/prep62/prep2011_62.pdf.
Yu. G. Rykov and O. B. Feodoritova, Preprint No. 84, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2014); http://keldysh.ru/papers/2014/prep2014_84.pdf.
E. Weinan, Yu. G. Rykov, and Ya. G. Sinai, “Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics,” Commun. Math. Phys. 177, 349–380 (1996).
A. I. Aptekarev and Yu. G. Rykov, “On the variational representation of solutions to some quasilinear equations and systems of hyperbolic type on the basis of potential theory,” Russ. J. Math. Phys. 13 (1), 4–12 (2006).
B. L. Keyfitz and H. C. Kranzer, “A viscous approximation to a system of conservation laws with no classical Riemann solution,” Lect. Notes Math. 1402, 185–197 (1989).
E. Hopf, “The partial differential equation u t + uu x = μuu x,” Commun. Pure Appl. Math. 3, 201–230 (1950).
O. A. Oleinik, “Cauchy problem for first-order nonlinear differential equations with discontinuous initial data,” Tr. Mosk. Mat. Ob–va 5, 433–454 (1956).
H. Nessyahu and E. Tadmor, “Nonoscillatory central differencing for hyperbolic conservation laws,” J. Comput. Phys. 87 (2), 408–463 (1990).
CentPack: A package of high-resolution central schemes for nonlinear conservation laws; http://www.cscamm.umd.edu/centpack.
T. A. Aleksandrikova and M. P. Galanin, Preprint No. 62, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2003); http://keldysh.ru/papers/2003/prep62/prep2003_62.html.
E. Tadmor, “Variational formulation of entropy solutions for nonlinear conservation laws,” Joint Math. Meeting, Baltimore, MD, January 2014; http://www.cscamm.umd.edu/tadmor/Lectures/2014%2001%20Variational_formulation_JMM_address%20printout.pdf.
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Original Russian Text © Yu.G. Rykov, O.B. Feodoritova, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 9, pp. 1586–1598.
In blessed memory of A.P. Favorskii
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Rykov, Y.G., Feodoritova, O.B. Systems of quasilinear conservation laws and algorithmization of variational principles. Comput. Math. and Math. Phys. 55, 1554–1566 (2015). https://doi.org/10.1134/S0965542515090122
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DOI: https://doi.org/10.1134/S0965542515090122