HLLC+: low-Mach shock-stable HLLC-type Riemann solver for all-speed flows.

*(English)*Zbl 1447.65045Authors’ abstract: The approximate Riemann solver of Harten-Lax-van Leer (HLL) and its variant HLLC (HLL with Contact restoration) solver are widely used as flux functions of finite volume Godunov-type methods for the solution of the gas dynamic Euler equations. However, the HLLC solver suffers from two significant difficulties: an accuracy problem at low-speed flows and shock instability at high-speed flows. To remedy such drawbacks, a novel low-Mach shock-stable HLLC-type scheme called HLLC\(+\) is developed for all speeds. The antidissipation pressure fix is introduced first to overcome the accuracy problem in low Mach number limits. Then, shear viscosity is identified and scaled into the original HLLC scheme to overcome shock instability. A new pressure-based factor function without switching coefficients is devised to prevent shear viscosity from smearing the boundary layer. The new HLLC\(+\) scheme involves no empirical parameters and is easy to implement. Asymptotic analysis and low Mach number test cases show the excellent behaviors of HLLC\(+\) in low Mach number limits: no global cut-off problem, damping pressure checkerboard modes, having expected \(\text{Ma}^2\) scaling of pressure and density fluctuations, and satisfaction of divergence constraint. Furthermore, this work manifests that the accuracy problem is associated with the normal velocity jumps of the flux interface, while shock instability is related to the transverse velocity jumps. Numerical test cases across a wide range of Mach numbers demonstrate the superior performance and potentiality of HLLC\(+\) to simulate all Mach number flows.

Reviewer: Victor Michel-Dansac (Strasbourg)

##### MSC:

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35C20 | Asymptotic expansions of solutions to PDEs |

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

76J20 | Supersonic flows |

76K05 | Hypersonic flows |

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

76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |

##### Keywords:

accuracy problem; shock instability; asymptotic analysis; all speeds; HLLC; computational fluid dynamics
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\textit{S. Chen} et al., SIAM J. Sci. Comput. 42, No. 4, B921--B950 (2020; Zbl 1447.65045)

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