Comparison among unstructured TVD, ENO and UNO schemes in two- and three-dimensions.

*(English)*Zbl 1426.76400Summary: This study focuses on unstructured TVD, ENO and UNO schemes applied to solve the Euler equations in two- and three-dimensions. They are implemented on a finite volume context and cell centered data base. The algorithms of H. C. Yee et al. [“A high-resolution numerical technique for inviscid gas-dynamic problems with weak solutions”, Lect. Notes Phys. 170, 546–552 (1982; doi:10.1007/3-540-11948-5_72)]; A. Harten [J. Comput. Phys. 49, 357–393 (1983; Zbl 0565.65050)]; H. C. Yee and P. Kutler [Application of second-order-accurate total variation diminishing (TVD) schemes to the Euler equations in general geometries. Techn. Rep., NASA Ames Research Center (1985)]; H. C. Yee, et al. [J. Comput. Phys. 57, 327–360 (1985; Zbl 0631.76087)]; H. C. Yee [J. Comput. Phys. 68, 151–179 (1987; Zbl 0621.76026)]; H. C. Yee and A. Harten [“Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates”, AIAA J. 25, No. 2, 266–274 (1987, doi:10.2514/3.9617)]; A. Harten and S. Osher [SIAM J. Numer. Anal. 24, 279–309 (1987; Zbl 0627.65102)]; J. Y. Yang [“Uniformly second-order-accurate essentially nonoscillatory schemes for the Euler equations”, AIAA J. 28, No. 12, 2069–2076 (1990; doi:10.2514/3.10523)], M. C. Hughson and P. S. Beran [“Analysis of hypersonic blunt-body flows using a total variation diminishing (TVD) scheme and the MacCormack scheme”, in: 9th Applied Aerodynamics Conference. Baltimore, MD: AIAA. (1991; doi:10.2514/6.1991-3206)]; J. Y. Yang [“Third-order nonoscillatory schemes for the Euler equations”, AIAA J. 29, No. 11, 1611-1618 (1991; doi:10.2514/3.10782)]; and J. Y. Yang and C. A. Hsu [AIAA J. 30, No. 6, 1570–1575 (1992; Zbl 0759.76054)] are implemented to solve such system of equations in two- and three-dimensions. All schemes are flux difference splitting and good resolution is expected. This study deals with calorically perfect gas model and in so on the cold gas formulation has been employed. Two problems are studied, namely: the transonic convergent-divergent symmetrical nozzle, and the supersonic ramp. A spatially variable time step is implemented to accelerate the convergence process. The results highlights the excellent performance of the J. Y. Yang [“Uniformly second-order-accurate essentially nonoscillatory schemes for the Euler equations” AIAA J. 28, No. 12, 2069–2076 (1990; doi:10.2514/3.10523)] TVD scheme, yielding an excellent pressure distribution at the two-dimensional nozzle wall, whereas the Harten and Osher scheme yields accurate values to the angle of the oblique shock wave and the best wall pressure distributions in the two-dimensional ramp problem. On the other hand, the excellent performance of the Harten scheme in the three-dimensional nozzle problem, yielding an excellent pressure distribution at the nozzle wall, and the Yee and Harten scheme yielding an accurate value to the angle of the oblique shock wave and the best wall pressure distribution in the three-dimensional ramp problem are of good quality.

##### MSC:

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

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

76Bxx | Incompressible inviscid fluids |

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

76L05 | Shock waves and blast waves in fluid mechanics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

Euler equations; finite volumes; transonic and supersonic flow; TVD; ENO and UNO schemes; two- and three-dimensional simulations; unstructured discretization
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\textit{E. S. de G. Maciel} and \textit{C. R. de Andrade}, Appl. Math. Comput. 321, 130--175 (2018; Zbl 1426.76400)

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##### References:

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