Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations.

*(English)*Zbl 1370.76116Summary: The numerical simulation of physical phenomena represented by non-linear hyperbolic systems of conservation laws presents specific difficulties mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp profiles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaption, which reduces the computational cost by using finer grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and programming difficulties. In this work we present a method obtained by the combination of a high-order shock capturing scheme, built from C.-W. Shu and S. Osher’s conservative formulation [J. Comput. Phys. 77, No. 2, 439–471 (1988; Zbl 0653.65072); ibid. 83, No. 1, 32–78 (1989; Zbl 0674.65061)], a fifth-order weighted essentially non-oscillatory (WENO) interpolatory technique [G.-S. Jiang and C.-W. Shu, ibid. 126, No. 1, 202–228 (1996; Zbl 0877.65065)] and R. Donat and A. Marquina’s flux-splitting method [ibid. 125, No. 1, 42–58 (1996; Zbl 0847.76049)], with the adaptive mesh refinement (AMR) technique of M. J. Berger and collaborators [with P. Colella, ibid. 82, No. 1, 64–84 (1989; Zbl 0665.76070); with J. Oliger, ibid. 53, 484–512 (1984; Zbl 0536.65071); M. J. Berger, Adaptive mesh refinement for hyperbolic partial differential equations. Stanford, CA: Stanford University. (PhD Thesis) (1982)].

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

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

##### Keywords:

hyperbolic systems of conservation laws; adaptive mesh refinement; Shu-Osher conservative scheme
PDF
BibTeX
XML
Cite

\textit{A. Baeza} and \textit{P. Mulet}, Int. J. Numer. Methods Fluids 52, No. 4, 455--471 (2006; Zbl 1370.76116)

Full Text:
DOI

##### References:

[1] | Harten, Journal of Computational Physics 50 pp 235– (1983) |

[2] | Osher, Mathematics of Computation 41 pp 321– (1983) |

[3] | Tan, Journal of Computational Physics 200 pp 347– (2004) |

[4] | Adaptive mesh refinement for hyperbolic partial differential equations. Ph.D. Thesis, Computer Science Department, Stanford University, 1982. |

[5] | Berger, Journal of Computational Physics 53 pp 484– (1984) |

[6] | Berger, Journal of Computational Physics 82 pp 64– (1989) |

[7] | An adaptive grid algorithm for computational shock hydrodynamics. Ph.D. Thesis, Cranfield Institute of Technology, 1991. |

[8] | Shu, Journal of Computational Physics 77 pp 439– (1988) |

[9] | Shu, Journal of Computational Physics 83 pp 32– (1989) |

[10] | Jiang, Journal of Computational Physics 126 pp 202– (1996) |

[11] | Donat, Journal of Computational Physics 125 pp 42– (1996) |

[12] | Marquina, Journal of Computational Physics 185 pp 120– (2003) |

[13] | Haas, Journal of Fluid Mechanics 181 pp 41– (1987) |

[14] | Donat, Journal of Computational Physics 146 pp 58– (1998) |

[15] | Liu, Journal of Computational Physics 115 pp 200– (1994) |

[16] | Quirk, Journal of Fluid Mechanics 318 pp 129– (1996) |

[17] | Chiavassa, SIAM Journal on Scientific Computing 23 pp 805– (2001) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.