# zbMATH — the first resource for mathematics

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.
Reviewer: R.H.W.Hoppe

##### MSC:
 76N15 Gas dynamics, general 76M99 Basic methods in fluid mechanics
Full Text:
##### References:
 [1] Bayliss, A; Turkel, E, Far field boundary conditions for compressible flows, () · Zbl 0494.76072 [2] Chorin, A.J, J. comput. phys., 23, 517, (1976) [3] Chorin, A.J, J. comput. phys., 25, 253, (1977) [4] Colella, P, A direct Eulerian MUSCL scheme for gas dynamics, () · Zbl 0562.76072 [5] Fok, S.K, Extension of Glimm’s method to the problem of gas flow in a duct of variable cross-section, () [6] Friedman, M.P, J. fluid mech., 8, 193, (1960) [7] Friedman, M.P, J. fluid mech., 11, 1, (1961) [8] Glimm, J, Comm. pure appl. math., 18, 697, (1965) [9] Harten, A, High resolution schemes for hyperbolic conservation laws, (1982), Preprint [10] Huang, L.C, J. comput. phys., 42, 195, (1981) [11] Liu, T.P, Comm. math. phys., 68, 141, (1979) [12] \scT. P. Liu, “Transonic Gas Flow in a Duct of Varying Area,” Arch. Rat. Mech. Anal., in press. · Zbl 0503.76076 [13] Marshall, G; Menendez, A.N, J. comput. phys., 44, 167, (1981) [14] Moretti, G, Thoughts and afterthoughts about shock computations, () [15] Moretti, G, Numerical analysis in gas dynamics, () · Zbl 0507.76012 [16] Shubin, G.R; Stephens, A.B; Glaz, H.M, J. comput. phys., 39, 364, (1981) [17] Sod, G.A, J. fluid mech., 83, 785, (1977) [18] Torres, J.A.D; Baker, R.C, Comput. fluids, 7, 177, (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.