A general-purpose finite-volume advection scheme for continuous and discontinuous fields on unstructured grids.

*(English)*Zbl 1143.76499Summary: We present a new general-purpose advection scheme for unstructured meshes based on the use of a variation of the interface-tracking flux formulation recently put forward by O. Ubbink and R. I. Issa [J. Comput. Phys. 153, No. 1, 26–50 (1999; Zbl 0955.76058)], in combination with an extended version of the flux-limited advection scheme of J. Thuburn [J. Comput. Phys. 123, No. 1, 74–83 (1996; Zbl 0840.76063)], for continuous fields. Thus, along with a high-order mode for continuous fields, the new scheme presented here includes optional integrated interface-tracking modes for discontinuous fields. In all modes, the method is conservative, monotonic, and compatible. It is also highly shape preserving. The scheme works on unstructured meshes composed of any kind of connectivity element, including triangular and quadrilateral elements in two dimensions and tetrahedral and hexahedral elements in three dimensions. The scheme is finite-volume based and is applicable to control-volume finite-element and edge-based node-centered computations. An explicit-implicit extension to the continuous-field scheme is provided only to allow for computations in which the local Courant number exceeds unity. The transition from the explicit mode to the implicit mode is performed locally and in a continuous fashion, providing a smooth hybrid explicit-implicit calculation. Results for a variety of test problems utilizing the continuous and discontinuous advection schemes are presented.

##### MSC:

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

76M20 | Finite difference methods applied to problems in fluid mechanics |

##### Software:

SLIC
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\textit{E. D. Dendy} et al., J. Comput. Phys. 180, No. 2, 559--583 (2002; Zbl 1143.76499)

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##### References:

[1] | Brown, P.N.; Saad, Y., Hybrid Krylov methods for non-linear systems of equations, SIAM J sci. stat. comput., 11, 450, (1990) · Zbl 0708.65049 |

[2] | Gaskell, H.; Lau, A.K.C., Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm, Int. J. numer. methods fluids, 8, 617, (1988) · Zbl 0668.76118 |

[3] | Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. comput. phys., 39, 201, (1981) · Zbl 0462.76020 |

[4] | Leonard, B.P., Stable and accurate convective modeling procedure based on quadratic upstream interpolation, Comput. methods appl. mech. eng., 19, 59, (1979) · Zbl 0423.76070 |

[5] | Leveque, R.J., High-resolution conservative algorithms for advection in incompressible flow, SIAM J. numer. anal., 33, 627, (1996) · Zbl 0852.76057 |

[6] | Noh, W.F.; Woodward, P., SLIC (simple line interface calculation), (1976), Springer-Verlag New York · Zbl 0382.76084 |

[7] | Rudman, M., Volume-tracking methods for interfacial flow calculations, Int. J. numer. methods fluids, 24, 671, (1997) · Zbl 0889.76069 |

[8] | Thuburn, J., Multidimensional flux-limited advection schemes, J. comput. phys., 123, 74, (1996) · Zbl 0840.76063 |

[9] | Ubbink, O., numerical prediction of two fluid systems with sharp interfaces, (1997), University of London |

[10] | Ubbink, O.; Issa, R.I., A method for capturing sharp interfaces on arbitrary meshes, J. comput. phys., 153, 26, (1999) · Zbl 0955.76058 |

[11] | D. L. Youngs, Time-dependent multi-material flow with large fluid distortion, in, Numerical Methods for Fluid Dynamics, edited by, K. W. Morton, and, M. J. Baines, Academic, New York, 1982, p, 273. |

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