Geodesic
Finite difference scheme of high order of convergence at a~nonstationary shock wave
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 2 (1999) no. 1, pp. 47-56

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The finite difference scheme is constructed for the hyperbolic system of two conservation laws of the “shallow water” theory. It has not less than the second order of weak convergence when calculating the nonstationar shock wave. This scheme is not monotone, however, as differentiated from all other known now “high accuracy” schemes (considering monotone ones), it reproduces the Hugoniot conditions with high accuracy and correspondingly conserves the high order of the strong local convergence in the area of the non-stationary shock influence. 
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     author = {V. V. Ostapenko},
     title = {Finite difference scheme of high order of convergence at a~nonstationary shock wave},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {47--56},
     publisher = {mathdoc},
     volume = {2},
     number = {1},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_1999_2_1_a4/}
}
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V. V. Ostapenko. Finite difference scheme of high order of convergence at a~nonstationary shock wave. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 2 (1999) no. 1, pp. 47-56. http://geodesic.mathdoc.fr/item/SJVM_1999_2_1_a4/