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**A generalized Riemann problem for quasi-one-dimensional gas flows.**
*(English)*
Zbl 0566.76056

The random choice method developed by the first author [Commun. Pure Appl. Math. 18, 697-715 (1965; Zbl 0141.289)] and numerically implemented by A. Chorin [J. Comput. Phys. 22, 517-533 (1976; Zbl 0354.65047)] has proven to be an efficient technique for computing solutions of one-dimensional flows described by homogeneous hyperbolic systems of conservation laws. Several generalizations of that method have been proposed to include inhomogeneous systems which typically arise when curvature effects cannot be neglected, but so far the applicability of these generalizations was limited to some extent [cf. e.g.: G. Sod, J. Fluid Mech. 83, 785-794 (1977; Zbl 0366.76055) and T. Liu, Commun. Math. Phys. 68, 141-172 (1979; Zbl 0435.35054)].

In the present paper the authors develop a generalized random choice method where for each time step a generalized Riemann problem (formed by two steady flows on adjacent spatial mesh intervals separated by a jump discontinuity) is solved with second order accuracy in time, and an approximate steady flow on the new time level is calculated by sampling the solution at a randomly chosen point. In practice, the computation of the generalized Riemann problem takes care of curvature and strengthening of waves but does not include secondary waves. At the end of the paper, results are given for transient gas flows in a Laval nozzle and compared with other numerical methods.

In the present paper the authors develop a generalized random choice method where for each time step a generalized Riemann problem (formed by two steady flows on adjacent spatial mesh intervals separated by a jump discontinuity) is solved with second order accuracy in time, and an approximate steady flow on the new time level is calculated by sampling the solution at a randomly chosen point. In practice, the computation of the generalized Riemann problem takes care of curvature and strengthening of waves but does not include secondary waves. At the end of the paper, results are given for transient gas flows in a Laval nozzle and compared with other numerical methods.

Reviewer: R.H.W.Hoppe

### Keywords:

hyperbolic systems of conservation laws; generalized random choice method; generalized Riemann problem; two steady flows; jump discontinuity; second order accuracy in time; transient gas flows in a Laval nozzle
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DOI

### References:

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