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), 1-29

MSC:
Primary 76H05; Secondary 65M05, 76-08

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771027-5

MathSciNet review:
771027

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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. 313-343.**[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. 363-372.**[3]**R. Chipman & A. Jameson,*Fully Conservative Numerical Solutions for Unsteady Irrotational Flow About Airfoils*, AIAA paper 79-1555, 1979.**[4]**P. Colella & P. R. Woodward,*The Piecewise-Parabolic Method*(*PPM*)*for Gas Dynamical Simulations*, LBL report #14661, July, 1982.**[5]**Ronald J. DiPerna,*Decay of solutions of hyperbolic systems of conservation laws with a convex extension*, Arch. Rational Mech. Anal.**64**(1977), no. 1, 1–46. MR**0454375**, https://doi.org/10.1007/BF00280177**[6]**A. Eberle,*Eine method finiter elements berechnung der transsonicken potential--strimung 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 Angle-of-Attack Effects*, AIAA Paper 82-0685, New Orleans, Louisiana, 1982.**[8]**Björn Engquist and Stanley Osher,*Stable and entropy satisfying approximations for transonic flow calculations*, Math. Comp.**34**(1980), no. 149, 45–75. MR**551290**, https://doi.org/10.1090/S0025-5718-1980-0551290-1**[9]**B. Engquist & S. Osher, "Discrete shocks and upwind schemes." (In preparation.)**[10]**K. O. Friedrichs and P. D. Lax,*Systems of conservation equations with a convex extension*, Proc. Nat. Acad. Sci. U.S.A.**68**(1971), 1686–1688. MR**0285799****[11]**S. K. Godunov,*A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics*, Mat. Sb. (N.S.)**47 (89)**(1959), 271–306 (Russian). MR**0119433****[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 83-0371, 1983.**[13]**P. M. Goorjian & R. van Buskirk,*Implicit Calculations of Transonic Flow Using Monotone Methods*, AIAA Paper 81-331, 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 78-1148, Seattle, Wash., 1978.**[15]**M. Hafez, W. Whitlow, Jr. & S. Osher, "Improved finite difference schemes for transonic potential calculations," AIAA Paper 84-0092, Reno, Nevada, 1984.**[16]**Ami Harten,*High resolution schemes for hyperbolic conservation laws*, J. Comput. Phys.**49**(1983), no. 3, 357–393. MR**701178**, https://doi.org/10.1016/0021-9991(83)90136-5**[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. 145-152.**[18]**Antony Jameson,*Numerical solution of nonlinear partial differential equations of mixed type*, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 275–320. MR**0468255****[19]**A. Jameson, "Transonic potential flow calculations using conservative form,"*AIAA Second Computational Fluid Dynamics Proceedings*, Hartford, Conn., 1975, pp. 148-155.**[20]**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****[21]**E. Murman, "Analysis of embedded Shockwaves calculated by relaxation methods,"*AIAA J.*, v. 12, 1974, pp. 626-633.**[22]**Stanley Osher,*Numerical solution of singular perturbation problems and hyperbolic systems of conservation laws*, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 179–204. MR**605507****[23]**Stanley Osher,*Riemann solvers, the entropy condition, and difference approximations*, SIAM J. Numer. Anal.**21**(1984), no. 2, 217–235. MR**736327**, https://doi.org/10.1137/0721016**[24]**Stanley Osher and Sukumar Chakravarthy,*High resolution schemes and the entropy condition*, SIAM J. Numer. Anal.**21**(1984), no. 5, 955–984. MR**760626**, https://doi.org/10.1137/0721060**[25]**P. L. Roe,*Approximate Riemann solvers, parameter vectors, and difference schemes*, J. Comput. Phys.**43**(1981), no. 2, 357–372. MR**640362**, https://doi.org/10.1016/0021-9991(81)90128-5**[26]**Ph. Morice & H. Vivand, "Équations de conservation et condition d'irréversibilité pour les écoulements transsoniques potentiels,"*C. R. Acad. Sci. Paris Sér. A-B*, v. 291, 1980, pp. B235-B238.**[27]**B. Van Leer, "Towards the ultimate conservative difference scheme, II. Monotonicity and conservation combined in a second-order scheme,"*J. Comput. Phys.*, v. 14, 1974, pp. 361-370.**[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. 101-136.**[29]**Bram van Leer,*On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe*, SIAM J. Sci. Statist. Comput.**5**(1984), no. 1, 1–20. MR**731878**, https://doi.org/10.1137/0905001**[30]**L. Carter Wellford Jr. and M. M. Hafez,*A finite element first-order equation formulation for the small-disturbance transonic flow problem*, Comput. Methods Appl. Mech. Engrg.**22**(1980), no. 2, 161–186. MR**577158**, https://doi.org/10.1016/0045-7825(80)90083-3**[31]**Peter Lax and Burton Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**0120774**, https://doi.org/10.1002/cpa.3160130205

Retrieve articles in *Mathematics of Computation*
with MSC:
76H05,
65M05,
76-08

Retrieve articles in all journals with MSC: 76H05, 65M05, 76-08

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771027-5

Keywords:
Full potential equation,
transonic flow,
entropy condition,
difference approximations

Article copyright:
© Copyright 1985
American Mathematical Society