Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On some numerical schemes for transonic flow problems

Author: Marco Mosché Mostrel
Journal: Math. Comp. 52 (1989), 587-613
MSC: Primary 65M05; Secondary 76H05
MathSciNet review: 955752
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: New second-order accurate finite difference approximations for a class of nonlinear PDE's of mixed type, which includes the 2D Low Frequency Transonic Small Disturbance equation (TSD) and the 2D Full Potential equation (FP), are presented.

For the TSD equation, the scheme is implemented via a time splitting algorithm; the inclusion of flux limiters keeps the total variation nonincreasing and eliminates spurious oscillations near shocks. Global Linear Stability, Total Variation Diminishing and Entropy Stability results are proven. Numerical results for the flow over a thin airfoil are presented. Current techniques used to solve the TSD equation may easily be extended to second-order accuracy by this method.

For the FP equation, the new scheme requires no subsonic/supersonic switching and no numerical flux biasing. Global Linear Stability for all values of the Mach number is proven.

References [Enhancements On Off] (What's this?)

  • [1] J. T. Batina, An Efficient Algorithm for Solution of the Unsteady Transonic Small-Disturbance Equation, AIAA Paper No. 87-0109, 1987.
  • [2] B. Engquist & S. Osher, "One-sided difference approximations for nonlinear conservation laws," Math. Comp., v. 6, 1981, pp. 321-351. MR 606500 (82c:65056)
  • [3] B. Engquist & S. Osher, "Stable and entropy satisfying approximations for transonic flow calculations," Math. Comp., v. 34, 1980, pp. 45-75. MR 551290 (81b:65082)
  • [4] K. O. Friedrichs & P. D. Lax, "Systems of conservation law equations with a convex extension," Proc. Nat. Acad. Sci. USA, v. 68, 1971, pp. 1686-1688. MR 0285799 (44:3016)
  • [5] 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. 271-306. (Russian) MR 0119433 (22:10194)
  • [6] P. M. Goorjian & R. Van Buskirk, Implicit Calculations of Transonic Flow Using Monotone Methods, AIAA Paper No. 81-0331, 1981.
  • [7] A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357-393. MR 701178 (84g:65115)
  • [8] S. N. Kružkov, "First order quasilinear equations in several independent variables," Math. USSR-Sb., v. 10, 1970, pp. 217-243.
  • [9] P. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217-237. MR 0120774 (22:11523)
  • [10] M. S. Mock, "Systems of conservation laws of mixed type," J. Differential Equations, v. 37, 1980, pp. 70-88. MR 583340 (81m:35088)
  • [11] M. M. Mostrel, On Some Numerical Schemes for Transonic Flow Problems, Ph.D. Thesis, University of California, Los Angeles, 1987.
  • [12] M. M. Mostrel, Second Order Accurate Finite Difference Approximations for the Transonic Small Disturbance Equation and the Full Potential Equation, Proceedings of the Winter Annual Meeting of the American Society of Mechanical Engineers, Chicago, IL, Nov. 27-Dec. 2, 1988, FED, volume 66 (O. Baysal, ed.), pp. 129-143.
  • [13] S. Osher, "Convergence of generalized MUSCL schemes," SIAM J. Numer. Anal., v. 22, 1985, pp. 947-961. MR 799122 (87b:65147)
  • [14] S. Osher, M. M. Hafez & W. Whitlow, Jr., "Entropy condition satisfying approximations for the full potential equation of transonic flow," Math. Comp., v. 44, 1985, pp. 1-29. MR 771027 (86c:76036)
  • [15] S. Osher, "Riemann solvers, the entropy condition, and difference approximations," SIAM J. Numer. Anal., v. 21, 1984, pp. 217-235. MR 736327 (86d:65119)
  • [16] D. A. Seidel & J. T. Batina, User's Manual for XTRAN2L (Version 1.2): A Program for Solving the General-Frequency Unsteady Transonic Small-Disturbance Equation, NASA Technical Memorandum No. 87737, 1986.
  • [17] V. Shankar, H. Ide, J. Gorski & S. Osher, A Fast, Time-Accurate Unsteady Full Potential Scheme, AIAA Paper No. 85-0165, 1985.
  • [18] V. Shankar, Implicit Treatment of the Unsteady Full Potential Equation in Conservation Form, AIAA Paper No. 84-0262, 1984.
  • [19] V. Shankar & S. Osher, "An efficient, full-potential implicit method based on characteristics for supersonic flows," AIAA J., v. 21, 1982, pp. 1262-1270. MR 714763 (85a:76066)
  • [20] 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.
  • [21] W. Whitlow, Jr., XTRAN2L: A Program for Solving the General-Frequency Unsteady Transonic Small Disturbance Equation, NASA Technical Memorandum No. 85723, 1983.

Similar Articles

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

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

Additional Information

Keywords: Full potential equation, transonic small disturbance equation, transonic flow, entropy condition, difference approximations
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society