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

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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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     author = {N. Ya. Moiseev and T. A. Mukhamadieva},
     title = {Newton's method as applied to the {Riemann} problem for media with general equations of state},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1102--1110},
     publisher = {mathdoc},
     volume = {48},
     number = {6},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_6_a12/}
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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/