Abstract
An approach based on Newton’s method is proposed for solving the Riemann problem for media with normal equations of state. The Riemann integrals are evaluated using a cubic approximation of an isentropic curve that is superior to the Simpson method in terms of accuracy, convergence rate, and efficiency. The potentials of the approach are demonstrated by solving problems for media obeying the Mie-Grüneisen equation of state. The algebraic equation of the isentropic curve and some exact solutions for configurations with rarefaction waves are explicitly given.
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References
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).
L. V. Ovsyannikov, Lectures on Basics of Gas Dynamics (Nauka, Moscow, 1981) [in Russian].
S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, et al., in Numerical Solution of Multidimensional Problems in Gas Dynamics (Nauka, Moscow, 1976) [in Russian].
G. B. Aladykin, S. K. Godunov, I. L. Kireeva, and L. A. Pliner, Solution of One-Dimensional Problems in Gas Dynamics on Moving Grids (Nauka, Moscow, 1970) [in Russian].
Yu. M. Shustov, “Solution to the Riemann Problem for an Arbitrary Equation of State,” in Numerical Methods in Continuum Mechanics (Vychisl. Tsentr Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1978), Vol. 9, No. 4, pp. 131–138 [in Russian].
V. F. Kuropatenko, G. V. Kovalenko, V. I. Kuznetsov, et al., “Software Package VOLNA and Nonhomogeneous Finite-Difference Method for Unsteady Motions of Compressible Continuum Media,” VANT, Ser. Metod. Program. Chisl. Reshen. Zadach Mat. Fiz. 2, 19–25 (1989).
A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman & Hall/CRC, London, 2001).
G. P. Prokopov, “Computation of a Discontinuity for Porous Media and Continuum Media with a Two-Term Equation of State,” VANT, Ser. Metodiki. Ser. Metod. Program. Chisl. Reshen. Zadach Mat. Fiz. 2(10), 32–40 (1982).
G. P. Prokopov, Preprint. No. 15, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2007).
T. A. Kobzeva and N. Ya. Moiseev, “Method of Undetermined Coefficients as Applied to the Riemann Problem,” VANT, Ser. Metod. Program. Chisl. Reshen. Zadach Mat. Fiz., 1, 3–9 (2003).
L. Collatz, Functional Analysis and Numerical Mathematics (Academic, New York, 1966; Mir, Moscow, 1969).
H. Weyl, “Shock Waves in Arbitrary Fluids,” Commun. Pure Appl. Math, 2, 103–122 (1949).
E. I. Zababakhin, Some Issues of Explosion Gas Dynamics (RFYaTs-VNIITF, Snezhinsk, 1997).
I. F. Kobylkin, V. V. Selivanov, V. S. Solov’ev, and N. N. Sysoev, Shock and Detonation Waves: Research Methods (Fizmatlit, Moscow, 2004) [in Russian].
H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, London, 1961; Nauka, Moscow, 1978).
I. N. Bronshtein and K. A. Semendyaev, Handbook of Mathematics for Engineers and Engineering Students (Nauka, Moscow, 1986) [in Russian].
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Original Russian Text © N.Ya. Moiseev, T.A. Mukhamadieva, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 6, pp. 1102–1110.
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Moiseev, N.Y., Mukhamadieva, T.A. Newton’s method as applied to the Riemann problem for media with general equations of state. Comput. Math. and Math. Phys. 48, 1039–1047 (2008). https://doi.org/10.1134/S0965542508060134
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DOI: https://doi.org/10.1134/S0965542508060134