A two-point difference scheme for computing steady-state solutions to the conservative one-dimensional Euler equations. (English) Zbl 0563.76023

An implicit finite-difference method is presented for obtaining steady- state solutions to the time dependent Euler equations, written in the conservation form, for flow containing shocks. The method uses a two- difference scheme for flux derivatives at subsonic points, at supersonic points, dissipation is added by retarded density concept, and the difference equations involve three points. However, the overall method retains the nice feature of the general two-point method as regards boundary conditions. Application of the method to quasi-one-dimensional nozzle flow equations is introduced for five different test cases, combining subsonic and supersonic boundary conditions. Some comparison with three-point method is made. Calculated results show the method’s advantages. Residuals can be reduced to machine zero in approximately 35 time steps for 50 mesh points. For one-dimensional Euler calculations, the scheme offers two advantages; the first is in regard to application of boundary conditions, the second is that the two-point algorithm is well-conditioned for large time steps.
Reviewer: G.C.Ling


76B99 Incompressible inviscid fluids
76M99 Basic methods in fluid mechanics
Full Text: DOI


[1] Jameson, A., Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method, (AIAA Paper 79-1458. AIAA Paper 79-1458, Proc. AIAA 4th Comp. Fld. Duns. Conf. (1979)), 122-146
[2] Moretti, G., Comment on stability aspects of diverging subsonic flows, AIAA J., 19, 5, 669 (1981)
[3] Cline, M. C., Stability aspects of diverging subsonic flow, AIAA J., 18, 5, 534-539 (1980)
[4] Yee, H. C.; Beam, R. M.; Warming, R. F., Stable boundary approximations for a class of implicit schemes for the one-dimensional inviscid equations of gas dynamics, (AIAA Paper 81-1009. AIAA Paper 81-1009, Proc. AIAA 5th Comp. Fld. Dyns. Conf. (1981)), 125-135 · Zbl 0496.76065
[5] Hafez, M.; South, J.; Murman, E., Artificial compressibility methods for numerical solutions of transonic full potential equation, AIAA J., 17, 8, 838-844 (1979) · Zbl 0409.76013
[6] Holst, T. L.; Ballhaus, W. F., Fast conservative schemes for the full potential equation applied to transonic flows, AIAA J., 17, 2, 145-152 (1979) · Zbl 0392.76045
[7] Blottner, F. G., (Influence of Physical and Computational Boundary Conditions on Quasi-One-Dimensional Flows (1977), Sandia Laboratories: Sandia Laboratories Albuquerque, New Mexico)
[8] Shubin, G. R.; Stephens, A. B.; Glaz, H. M., Steady shock tracking and Newton’s method applied to one-dimensional duct flow, J. Comp. Phys., 39, 364-374 (1981) · Zbl 0468.76061
[9] Wornom, S., Implicit characteristic modelling schemes for the Euler equations—A new approach, (Presented at the 6th AIAA Comp. Conf.. Presented at the 6th AIAA Comp. Conf., Danvers, Massachusetts (1983))
[10] Bredif, M., A fast finite element method for transonic potential flow calculations, (AIAA Preprint 83-0507. AIAA Preprint 83-0507, Reno, Nevada. AIAA Preprint 83-0507. AIAA Preprint 83-0507, Reno, Nevada, 21st Aero-space Meeting (Jan. 1918)) · Zbl 0554.76016
[11] Phillips, R. B.; Rose, M. E., Compact difference schemes for mixed initial boundary value problems—I, ICASE Report No. 81-4 (21 Jan. 1981)
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.