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**A sequel to AUSM II: AUSM\(^+\)-up for all speeds.**
*(English)*
Zbl 1137.76344

Summary: We present ideas and procedure to extend the AUSM-family schemes to solve flows at all speed regimes. To achieve this, we first focus on the theoretical development for the low Mach number limit. Specifically, we employ asymptotic analysis to formally derive proper scalings for the numerical fluxes in the limit of small Mach number. The resulting new scheme is shown to be simple and remarkably improved from previous schemes in robustness and accuracy. The convergence rate is shown to be independent of Mach number in the low Mach number regime up to \(M_{\infty} = 0.5\), and it is also essentially constant in the transonic and supersonic regimes. Contrary to previous findings, the solution remains stable, even if no local preconditioning matrix is included in the time derivative term, albeit a different convergence history may occur. Moreover, the new scheme is demonstrated to be accurate against analytical and experimental results. In summary, the new scheme, named AUSM+-up, improves over previous versions and eradicates fails found therein.

### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M12 | Finite volume methods applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

### Keywords:

Upwind scheme; Low Mach number; AUSM scheme; AUSM\(^{+}\)-up; Positivity; Entropy-satisfying; Carbuncle phenomenon; Euler and Navier-Stokes equations
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\textit{M.-S. Liou}, J. Comput. Phys. 214, No. 1, 137--170 (2006; Zbl 1137.76344)

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### References:

[1] | Turkel, E., Preconditioned methods for solving incompressible and low speed compressible equations, Journal of computational physics, 72, 277-298, (1987) · Zbl 0633.76069 |

[2] | Choi, Y.H.; Merkle, C.L., The application of preconditioning in viscous flows, Journal of computational physics, 105, 207-223, (1993) · Zbl 0768.76032 |

[3] | J.M. Weiss, W.A. Smith, Preconditioning applied to variable and constant density time-accurate flows on unstructured meshes, AIAA Paper 94-2209, June 1994. |

[4] | B. van Leer, W.T. Lee, P.L. Roe, Characteristic time stepping or local preconditioning of the Euler equations, in: 10th AIAA CFD Conference, Paper 91-1552-CP, 1991. |

[5] | Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of computational physics, 43, 357-372, (1981) · Zbl 0474.65066 |

[6] | R.C. Chima, M.-S. Liou, Comparison of the AUSM^{+} and H-CUSP schemes for turbomachinery applications, AIAA Paper 2003-4120-CP, in: 16th AIAA CFD Conference, June 2003. |

[7] | Edwards, J.R.; Liou, M.-S., Low-diffusion flux-splitting methods for flows at all speeds, AIAA journal, 36, 1610-1617, (1998) |

[8] | M.-S. Liou, J.R. Edwards, Numerical speed of sound and its application to schemes for all speeds, AIAA Paper 99-3268-CP, in: 14th AIAA CFD Conference, 1999. |

[9] | M.-S. Liou, P.G. Buning, Contribution of the recent AUSM schemes to the OVERFLOW code: implementation and validation, AIAA Paper 2000-4404, 2000. |

[10] | Mary, I.; Sagaut, P.; Deville, M., An algorithm for unsteady viscous flows at all speeds, International journal of numerical methods in fluids, 34, 371-401, (2000) · Zbl 1003.76057 |

[11] | Edwards, J.R.; Franklin, R.K.; Liou, M.-S., Low-diffusion flux-splitting methods for real fluid flows at all speeds, AIAA journal, 38, 1624-1633, (2000) |

[12] | J. Edwards, D. Mao, Development of low-diffusion flux-splitting methods for dense gas-solid flows, AIAA Paper 2001-2649-CP, 2001. |

[13] | Paillère, H.; Core, C.; Garcia, J., On the extension of the AUSM^{+} scheme to compressible two-fluid models, Computers & fluids, 32, 891-916, (2003) · Zbl 1040.76044 |

[14] | C.-H. Chang, M.-S. Liou, A new approach to the simulation of compressible multifluid flows with AUSM^{+} scheme, AIAA Paper 2003-4107, 2003. |

[15] | Liou, M.-S.; Steffen, C.J., A new flux splitting scheme, Journal of computational physics, 107, 23-39, (1993), Also NASA TM 104404, May 1991 · Zbl 0779.76056 |

[16] | Liou, M.-S., A sequel to AUSM: AUSM^{+}, Journal of computational physics, 129, 364-382, (1996), Also NASA TM 106524, March 1994 · Zbl 0870.76049 |

[17] | Kim, K.H.; Kim, C.; Rho, O., Methods for the accurate computations of hypersonic flows I. AUSMPW^{+} scheme, Journal of computational physics, 174, 38-80, (2001) · Zbl 1106.76421 |

[18] | B. Müller, Low mach number asymptotics of the Navier-Stokes equations and numerical implications, VKI CFD Lecture Series, 1999-03, 1999. |

[19] | Rehm, R.G.; Baum, H.R., The equations of motion for thermally driven buoyant flows, Journal of research of the national bureau of standards, 83, 297-308, (1978) · Zbl 0433.76072 |

[20] | M.-S. Liou, Ten years in the making-AUSM-family, AIAA Paper 2001-2521-CP, in: 15th AIAA CFD Conference, June 11-14, 2001. |

[21] | Van Leer, B., Flux-vector splitting for the Euler equations, Lecture notes in physics, 170, 507-512, (1982) |

[22] | Roberts, T.W., The behavior of flux difference splitting schemes near slowly moving shock waves, Journal of computational physics, 90, 141-160, (1990) · Zbl 0699.76076 |

[23] | Godunov, S.K., A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations, Mat. sb, 47, 271-306, (1959) · Zbl 0171.46204 |

[24] | Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM journal on numerical analysis, 25, 294-318, (1988) · Zbl 0642.76088 |

[25] | Wada, Y.; Liou, M.-S., An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM journal on scientific and statistical computing, 18, 633-657, (1997) · Zbl 0879.76064 |

[26] | Liou, M.-S., Mass flux schemes and connection to shock instability, Journal of computational physics, 160, 623-648, (2000) · Zbl 0967.76062 |

[27] | Quirk, J.J., A contribution to the great Riemann solver debate, International journal of numerical methods fluids, 18, 555-574, (1994), Also ICASE Report 92-64, 1992 · Zbl 0794.76061 |

[28] | P.G. Buning, et. al., OVERFLOW User’s Manual, version 1.8f, Unpublished NASA Report, 1998. |

[29] | R.V. Chima, Calculation of multistage turbomachinery using steady characteristic boundary conditions, AIAA Paper 98-0968. Also NASA TM-1998-206613, 1998. |

[30] | Mentor, F.R., Two-equation eddy-viscosity turbulence models for engineering applications, AIAA journal, 32, 1598-1605, (1994) |

[31] | Wilcox, D.C., Reassessment of the scale-determining equation for advanced turbulence models, AIAA journal, 26, 1299-1310, (1988) · Zbl 0664.76057 |

[32] | P.R. Spalart, S.R. Allmaras, A one-equation turbulence model for aerodynamic flows, AIAA Paper 92-0439, 1992. |

[33] | Koren, B., Upwind schemes, multigrid and defect correction for the steady Navier-Stokes equations, Lecture notes in physics, 323, 344-348, (1989) |

[34] | Bachalo, W.D.; Johnson, D.A., Transonic turbulent boundary-layer separation generated on an axisymmetric flow model, AIAA journal, 24, 437-443, (1986) |

[35] | Vienrendeels, J.; Merci, B.; Dick, E., Blended AUSM^{+} method for all speeds and all grid aspect ratios, AIAA journal, 39, 2278-2282, (2001) |

[36] | L.J. Goldman, R.G. Seasholtz, Laser anemometer measurements in an annular cascade of core turbine vanes and comparison with theory, NASA TP 2018, NASA Lewis Research Center, 1982. |

[37] | P.L. Johnson, K.M. Jones, M.D. Madson, Experimental investigation of a simplified 3D high lift configuration in support of CFD validation, AIAA Paper 2000-4217, 2000. |

[38] | A. Jameson, W. Schmidt, E. Turkel, Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time stepping, AIAA Paper 81-1259, 1981. |

[39] | D.A. Treadgold, A.F. Jones, K.H. Wilson, Pressure distribution measurement in the RA 8ft×6ft transonic wind tunnel on RAE wing “A” in combination with an axi-symmetric body at mach numbers of 0.4, 0.8 and 0.9, Appendix B4, AGARD-AR-138, 1984. |

[40] | S.E. Rogers, K. Roth, S.M. Nash, CFD validation of high-lift flows with significant wind-tunnel effects, AIAA Paper 2000-4218-CP, in: AIAA 18th Applied Aerodynamics Conference, Denver, CO, 2000. |

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