A rotationally biased upwind difference scheme for the Euler equations. (English) Zbl 0557.76067

Summary: The upwind difference schemes of Godunov, Osher, Roe and van Leer are able to resolve one-dimensional steady shocks for the Euler equations within one or two mesh intervals. Unfortunately, this resolution is lost in two dimensions when the shock crosses the computing grid at an oblique angle. To correct this problem, a numerical scheme is developed which automatically locates the angle at which a shock might be expected to cross the computing grid and then constructs separate finite difference formulas for the flux components normal and tangential to this direction. Numerical results are presented which illustrate the ability of this new method to resolve steady oblique shocks.


76L05 Shock waves and blast waves in fluid mechanics
76M99 Basic methods in fluid mechanics
76N15 Gas dynamics (general theory)


Full Text: DOI


[1] Chakravarthy, S.; Osher, S., Numerical experiments with the osher upwind scheme for the Euler equations, () · Zbl 0526.76074
[2] Godunov, S.K., A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, (1959), also Cornell Aeronautical Lab (CALSPAN) translation · Zbl 0171.46204
[3] Harten, A.; Hyman, J.M., A self-adjusting grid for the computation of weak solutions of hyperbolic conservation laws, ()
[4] Harten, A.; Lax, P.D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35-62, (1983) · Zbl 0565.65051
[5] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357-393, (1983) · Zbl 0565.65050
[6] Jameson, A., Iterative solution of transonic flows over airfoils and wings, including flows at Mach 1, Comm. pure appl. math., 27, 283-309, (1974) · Zbl 0296.76033
[7] Lax, P.; Wendroff, B., Systems of conservation laws, Comm. pure appl. math., 13, 217-237, (1960) · Zbl 0152.44802
[8] Lax, P.D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, (1972), SIAM Philadelphia
[9] Liepmann, H.W.; Roshko, A., Elements of gasynamics, (1957), Wiley New York · Zbl 0078.39901
[10] Murman, E.M., Analysis of embedded shock waves calculated by relaxation methods, Aiaa j., 12, 626-633, (1974) · Zbl 0282.76064
[11] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. comp., 38, 339-374, (1982) · Zbl 0483.65055
[12] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[13] Segal, L.A., Mathematics applied to continuum mechanics, (1977), Macmillan Co New York
[14] Van Albada, G.D.; Van Leer, B.; Roberts, W.W., A comparative study of computational methods in cosmic gas dynamics, ICASE report 81-24, (1981)
[15] Van Leer, B., On the relation between the upwind-differencing schemes of Godunov, enquist-osher and roe, ICASE report 81-11, (1981)
[16] Van Leer, B., Flux vector splitting for the Euler equations, ICASE report 82-30, (1982)
[17] Woodward, P.; Colella, P., ()
[18] Yanenko, N.N., The method of fractional steps, (1971), Springer-Verlag New York/Berlin · Zbl 0209.47103
[19] Yee, H.C.; Warning, R.F.; Harten, A., On the application and extension of Harten’s high resolution scheme, Nasa tm 84256, (1982)
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