Entropy condition satisfying approximations for the full potential equation of transonic flow
Authors:
Stanley Osher, Mohamed Hafez and Woodrow Whitlow
Journal:
Math. Comp. 44 (1985), 129
MSC:
Primary 76H05; Secondary 65M05, 7608
MathSciNet review:
771027
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Abstract: We shall present a new class of conservative difference approximations for the steady full potential equation. They are, in general, easier to program than the usual density biasing algorithms, and in fact, differ only slightly from them. We prove rigorously that these new schemes satisfy a new discrete "entropy inequality", which rules out expansion shocks, and that they have sharp, steady, discrete shocks. A key tool in our analysis is the construction of an "entropy inequality" for the full potential equation itself. We conclude by presenting results of some numerical experiments using our new schemes.
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 [1]
 J. W. Boerstoel, "A multigrid algorithm for steady transonic potential flows around airfoils using Newton's iteration," J. Comput. Phys., v. 48, 1982, pp. 313343.
 [2]
 S. R. Chakravarthy & S. Osher, "High resolution applications of the Osher upwind scheme for the Euler equations," AIAA Computational Fluid Dynamics Proceedings, Danvers, Mass., 1983, pp. 363372.
 [3]
 R. Chipman & A. Jameson, Fully Conservative Numerical Solutions for Unsteady Irrotational Flow About Airfoils, AIAA paper 791555, 1979.
 [4]
 P. Colella & P. R. Woodward, The PiecewiseParabolic Method (PPM) for Gas Dynamical Simulations, LBL report #14661, July, 1982.
 [5]
 R. J. DiPerna, "Decay of solutions of hyperbolic systems of conservation laws with a convex extension," Arch. Rational Mech. Anal., v. 64, 1977, pp. 192. MR 0454375 (56:12626)
 [6]
 A. Eberle, Eine method finiter elements berechnung der transsonicken potentialstrimung un profile, MBB Berech Nr. UFE 1352(0), 1977.
 [7]
 J. W. Edwards, R. M. Bennett, W. Whitlow, Jr. & D. A. Seidel, Time Marching Transonic Flutter Solutions Including AngleofAttack Effects, AIAA Paper 820685, New Orleans, Louisiana, 1982.
 [8]
 B. Engquist & S. Osher, "Stable and entropy satisfying approximations for transonic flow calculations," Math. Comp., v. 34, 1980, pp. 4575. MR 551290 (81b:65082)
 [9]
 B. Engquist & S. Osher, "Discrete shocks and upwind schemes." (In preparation.)
 [10]
 K. O. Friedrichs & P. D. Lax, "Systems of conservation laws with a convex extension," Proc. Nat. Acad. Sci. U.S.A., v. 68, 1971, pp. 16861688. MR 0285799 (44:3016)
 [11]
 S. K. Godunov, "A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics," Mat. Sb., v. 47, 1959, pp. 271290. (Russian) MR 0119433 (22:10194)
 [12]
 P. M. Goorjian, M. Meagher & R. van Buskirk, Monotone Implicit Algorithms for the Small Disturbance and Full Potential Equation Applied to Transonic Flow, AIAA paper 830371, 1983.
 [13]
 P. M. Goorjian & R. van Buskirk, Implicit Calculations of Transonic Flow Using Monotone Methods, AIAA Paper 81331, St. Louis, Missouri, 1981.
 [14]
 M. M. Hafez, E. M. Murman & J. E. South, Artificial Compressibility Methods for Numerical Solution of Transonic Full Potential Equation, AIAA paper 781148, Seattle, Wash., 1978.
 [15]
 M. Hafez, W. Whitlow, Jr. & S. Osher, "Improved finite difference schemes for transonic potential calculations," AIAA Paper 840092, Reno, Nevada, 1984.
 [16]
 A. Harten, "High resolution schemes for conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [17]
 T. L. Holst & W. F. Ballhaus, "Fast, conservative schemes for the full potential equation applied to transonic flows," AIAA J., v. 17, 1979, pp. 145152.
 [18]
 A. Jameson, "Numerical solutions of nonlinear partial differential equations of mixed type," Numerical Solutions of Partial Differential Equations III, Academic Press, New York, 1976, pp. 275320. MR 0468255 (57:8093)
 [19]
 A. Jameson, "Transonic potential flow calculations using conservative form," AIAA Second Computational Fluid Dynamics Proceedings, Hartford, Conn., 1975, pp. 148155.
 [20]
 P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conf. Series Lectures in Appl. Math., Vol. 11, 1972. MR 0350216 (50:2709)
 [21]
 E. Murman, "Analysis of embedded Shockwaves calculated by relaxation methods," AIAA J., v. 12, 1974, pp. 626633.
 [22]
 S. Osher, Numerical Solution of Singular Perturbation Problems and Hyperbolic Systems of Conservation Laws, NorthHolland Math. Stud. No. 47 (O. Axelsson, L. S. Frank and A. van der Sluis, eds.), 1981, pp. 179205. MR 605507 (83g:65098)
 [23]
 S. Osher, "Riemann solvers, the entropy condition, and difference approximations," SIAM J. Numer. Anal., v. 21, 1984, pp. 217235. MR 736327 (86d:65119)
 [24]
 S. Osher & S. Chakravarthy, "High resolution schemes and the entropy condition," SIAM J. Numer. Anal., v. 21, 1984, pp. 955984. MR 760626 (86a:65086)
 [25]
 P. L. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes," J. Comput. Phys., v. 43, 1981, pp. 357372. MR 640362 (82k:65055)
 [26]
 Ph. Morice & H. Vivand, "Équations de conservation et condition d'irréversibilité pour les écoulements transsoniques potentiels," C. R. Acad. Sci. Paris Sér. AB, v. 291, 1980, pp. B235B238.
 [27]
 B. Van Leer, "Towards the ultimate conservative difference scheme, II. Monotonicity and conservation combined in a secondorder scheme," J. Comput. Phys., v. 14, 1974, pp. 361370.
 [28]
 B. Van Leer, "Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method," J. Comput. Phys., v. 32, 1979, pp. 101136.
 [29]
 B. Van Leer, "On the relation between the upwinddifferencing schemes of Godunov, EngquistOsher, and Roe," SIAM J. Sci. Statist. Comput., v. 5, 1984, pp. 120. MR 731878 (86a:65085)
 [30]
 L. C. Wellford, Jr. & M. M. Hafez, "A finite element firstorder equation for the small disturbance transonic flow problem," Comput. Methods Appl. Mech. Engrg., v. 22, 1980, pp. 161186. MR 577158 (81d:76067)
 [31]
 P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 23, 1960, pp. 217237. MR 0120774 (22:11523)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507710275
PII:
S 00255718(1985)07710275
Keywords:
Full potential equation,
transonic flow,
entropy condition,
difference approximations
Article copyright:
© Copyright 1985
American Mathematical Society
