Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids.

*(English)*Zbl 0941.65082The main purpose of the paper is the presentation of a new central essentially nonoscillatory (ENO) scheme, which is of third order and does not require field-by-field decomposition. First, the standard ENO schemes developed by C.-W. Shu and S. Osher [J. Comput. Phys. 83, No. 1, 32-78 (1989; Zbl 0674.65061)] are revisited by using point values instead of cell averages, which avoids staggering, as well as component-wise limiting. Next the authors introduce a new type of decision process called “convex ENO”, which yields the desired scheme. They insist on the fact that their flux does degenerate to first order at discontinuities. Furthermore, their scheme is easily extended to multi-dimensions. Numerical results are reported in detail for 10 significant test problems in one or two dimensions. They are excellent. It is also pointed out that the component-wise calculation is twice as fast as the field-by-field decomposition version of the scheme in each dimension.

Reviewer: S.Benzoni (Lyon)

##### MSC:

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

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

##### Keywords:

central schemes; essentially nonoscillatory scheme; component-wise calculation; high resolution methods; multi-dimensions; numerical results; ENO scheme
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\textit{X.-D. Liu} and \textit{S. Osher}, J. Comput. Phys. 142, No. 2, 304--330 (1998; Zbl 0941.65082)

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##### References:

[1] | Arora, M.; Roe, P.L., A well-behaved TVD limiter for high-resolution calculations of unsteady flow, J. comput. phys., 132, 3, (1997) · Zbl 0878.76045 |

[2] | Hwajeong, Choi, Jian-Guo, Liu, The reconstruction of upwind fluxes for conservation laws · Zbl 0935.76048 |

[3] | Engquist, B.; Runborg, O., Multi-phase computations in geometrical optics, Comput. appl. math., 74, 175, (1996) · Zbl 0947.78001 |

[4] | Harten, A., On a class of high resolution total-variation-stable finite-difference schemes, Sinum, 21, 1, (1984) · Zbl 0547.65062 |

[5] | Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357, (1983) · Zbl 0565.65050 |

[6] | Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes, III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067 |

[7] | Harten, A.; Osher, S., Uniformly high order accurate non-oscillatory scheme, Sinum, 24, 229, (1982) |

[8] | G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher, E. Tadmor, High resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation laws, SINUM · Zbl 0920.65053 |

[9] | Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202, (1996) · Zbl 0877.65065 |

[10] | G.-S. Jiang, E. Tadmor, Non-oscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws · Zbl 0914.65095 |

[11] | Jin, S.; Xin, Z., The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. pure appl. math., 48, 235, (1995) · Zbl 0826.65078 |

[12] | Liu, X.D.; Osher, S., Non-oscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes, I, Sinum, 33, (1996) · Zbl 0859.65091 |

[13] | X. D. Liu, E. Tadmor, Third order non-oscillatory central scheme for hyperbolic conservation laws, Numer. Math. · Zbl 0906.65093 |

[14] | E. Morano, R. Sanders, M.-C. Druguet, Multi-dimensional dissipation for upwind schemes: Stablity and applications to gas dynamics · Zbl 0924.76076 |

[15] | Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. comput. phys., 87, 408, (1990) · Zbl 0697.65068 |

[16] | Osher, S., Riemann solvers, the entropy condition and difference approximations, Sinum, 21, 955, (1984) |

[17] | Shu, C.-W., Numerical experiments on the accuracy of ENO and modified ENO schemes, J. sci. comput., 5, 127, (1990) · Zbl 0732.65085 |

[18] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072 |

[19] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061 |

[20] | Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, Sinum, 21, 995, (1984) · Zbl 0565.65048 |

[21] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115, (1984) · Zbl 0573.76057 |

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