Geodesic
Two-stages marching method for calculation viscous flow through Laval nozzle
Matematičeskoe modelirovanie, Tome 11 (1999) no. 7, pp. 95-117.

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Two-stages marching method is developed for computing viscous internal flows with subsonic and supersonic regions on the basis of simplified Navier-Stokes parabolic equations named smooth channel approximation. On the first stage the full region of subsonic flow is computed by marching method I untfer given flow rate till critical. The critical flow rate is calculated with controlled accuracy on the given numerical mesh. On the second stage the regularization of ill-posed equations is proposed, and the critical field flow through Laval nozzle is calculated up to end. The Petuchov's numerical scheme of high order consistency is generalized for computing the system of equations of second and first order simultaneously. The convergence is investigated under different Reynolds numbers. The marching step is derived. On the second stage the step by arithmetic progression is obtained in terms of Reynolds number. 
@article{MM_1999_11_7_a8,
     author = {N. N. Kalitkin and B. V. Rogov and I. A. Sokolova},
     title = {Two-stages marching method for calculation viscous flow through {Laval} nozzle},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {95--117},
     publisher = {mathdoc},
     volume = {11},
     number = {7},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_1999_11_7_a8/}
}
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N. N. Kalitkin; B. V. Rogov; I. A. Sokolova. Two-stages marching method for calculation viscous flow through Laval nozzle. Matematičeskoe modelirovanie, Tome 11 (1999) no. 7, pp. 95-117. http://geodesic.mathdoc.fr/item/MM_1999_11_7_a8/