Geodesic
Method for dynamic adaptation in problems of gas dynamiks
Matematičeskoe modelirovanie, Tome 7 (1995) no. 12, pp. 48-78.

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Application of a method for dynamic adaptation to numerical solution of unsteady-state hyperbolic equations is considered. The basic for given method is a transition to arbitrary nonstationary coordinate system. This allows to formulate a unified difference model to determine both numerical solution and grid movement. Two different techniques for computation of shock waves by means of dynamic adaptation are examined by the example of numerical solution for a model problem which describes accelerating piston motion in gas. The first technique enables the discontinuity surface to show up as a region of high gradients which grid nodes are concentrated in. The second one permits to separate the discontinuity explicitly by means of Rankine–Hugoniot equations. Both of these techniques don't use artificial viscosity and allow to proceed computations when using grids with small number of nodes ($N\approx10-20$). 
@article{MM_1995_7_12_a3,
     author = {P. V. Breslavskiy and V. I. Mazhukin},
     title = {Method for dynamic adaptation in problems of gas dynamiks},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {48--78},
     publisher = {mathdoc},
     volume = {7},
     number = {12},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_1995_7_12_a3/}
}
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P. V. Breslavskiy; V. I. Mazhukin. Method for dynamic adaptation in problems of gas dynamiks. Matematičeskoe modelirovanie, Tome 7 (1995) no. 12, pp. 48-78. http://geodesic.mathdoc.fr/item/MM_1995_7_12_a3/