Geodesic
Approximation of Hugoniot's conditions by explicit conservative difference schemes for non-stationar shock waves
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 1, pp. 77-88.

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Introducted here, is the concept of ($\varepsilon,\delta$)-Hugoniot's condition being the relatioship which links generalised solution magnitudes in points $(t-\delta,x(t)+\varepsilon)$ and $(t+\delta,x(t)-\varepsilon)$ for both sides of non-stationary shock wave front line $x=x(t)$. It is showed here, that the explicit bi-layer with respect to time conservative difference schemes for $\delta\ne0$ approximate ($\varepsilon,\delta$)-Hugoniot's conditions only with the first order, independent of their accuracy for smooth solutions. At the same time, if the front lines are quite smooth, then for $\delta=0$ these schemes approximate ($\varepsilon,0$)-Hugoniot's conditions with the same order they have for smooth solutions. 
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V. V. Ostapenko. Approximation of Hugoniot's conditions by explicit conservative difference schemes for non-stationar shock waves. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 1, pp. 77-88. http://geodesic.mathdoc.fr/item/SJVM_1998_1_1_a6/

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