On some numerical schemes for transonic flow problems

Author:
Marco Mosché Mostrel

Journal:
Math. Comp. **52** (1989), 587-613

MSC:
Primary 65M05; Secondary 76H05

DOI:
https://doi.org/10.1090/S0025-5718-1989-0955752-2

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.

**[1]**J. T. Batina,*An Efficient Algorithm for Solution of the Unsteady Transonic Small-Disturbance Equation*, AIAA Paper No. 87-0109, 1987.**[2]**Björn Engquist and Stanley Osher,*One-sided difference approximations for nonlinear conservation laws*, Math. Comp.**36**(1981), no. 154, 321–351. MR**606500**, https://doi.org/10.1090/S0025-5718-1981-0606500-X**[3]**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**[4]**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****[5]**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****[6]**P. M. Goorjian & R. Van Buskirk,*Implicit Calculations of Transonic Flow Using Monotone Methods*, AIAA Paper No. 81-0331, 1981.**[7]**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**[8]**S. N. Kružkov, "First order quasilinear equations in several independent variables,"*Math. USSR-Sb.*, v. 10, 1970, pp. 217-243.**[9]**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**[10]**M. S. Mock,*Systems of conservation laws of mixed type*, J. Differential Equations**37**(1980), no. 1, 70–88. MR**583340**, https://doi.org/10.1016/0022-0396(80)90089-3**[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]**Stanley Osher,*Convergence of generalized MUSCL schemes*, SIAM J. Numer. Anal.**22**(1985), no. 5, 947–961. MR**799122**, https://doi.org/10.1137/0722057**[14]**Stanley Osher, Mohamed Hafez, and Woodrow Whitlow Jr.,*Entropy condition satisfying approximations for the full potential equation of transonic flow*, Math. Comp.**44**(1985), no. 169, 1–29. MR**771027**, https://doi.org/10.1090/S0025-5718-1985-0771027-5**[15]**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**[16]**D. A. Seidel & J. T. Batina,*User's Manual for XTRAN*2*L*(*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]**Vijaya Shankar and Stanley Osher,*An efficient, full-potential implicit method based on characteristics for supersonic flows*, AIAA J.**21**(1983), no. 9, 1262–1270. MR**714763**, https://doi.org/10.2514/3.8238**[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.,*XTRAN*2*L*:*A Program for Solving the General-Frequency Unsteady Transonic Small Disturbance Equation*, NASA Technical Memorandum No. 85723, 1983.

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0955752-2

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

Article copyright:
© Copyright 1989
American Mathematical Society