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

*(English)*Zbl 1137.76344Summary: 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 |

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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