Geodesic
Newton's method as applied to the Riemann problem for media with general equations of state
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 6, pp. 1102-1110 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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N. Ya. Moiseev; T. A. Mukhamadieva. Newton's method as applied to the Riemann problem for media with general equations of state. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 6, pp. 1102-1110. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_6_a12/

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