Least-squares finite element methods for compressible Euler equations.

*(English)*Zbl 0692.76068Summary: A method based on backward finite differencing in time and a least- squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the \(L^ 2\)-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the \(L^ 2\)-method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted \(H^ 1\)-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

##### MSC:

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

65N99 | Numerical methods for partial differential equations, boundary value problems |

76M99 | Basic methods in fluid mechanics |

##### Keywords:

backward finite differencing; least-squares finite element scheme; Euler equations for gas dynamics; numerical dissipation; conservative least- squares finite element method
PDF
BibTeX
XML
Cite

\textit{B.-N. Jiang} and \textit{G. F. Carey}, Int. J. Numer. Methods Fluids 10, No. 5, 557--568 (1990; Zbl 0692.76068)

Full Text:
DOI

##### References:

[1] | Hughes, Int. j. numer. methods fluids 77 pp 1261– (1987) |

[2] | Donea, J. Comput. Phys. 70 pp 463– (1987) |

[3] | and , ’Finite element schemes for inviscid compressible flows’, Trans. 8th Int. Conf. on Structural Mechanics in Reactor Technology, Vol. B, North-Holland, 1985, pp. 111-115. |

[4] | Baker, Int. j. numer. methods fluids 7 pp 1235– (1987) |

[5] | Oden, Int. j. numer. methods fluids 7 pp 1211– (1987) |

[6] | LĂ¶hner, Int. j. numer. methods fluids 7 pp 93– (1987) |

[7] | and , ’Convection dominated problems’, in and (eds), Numerical Methods for Compressible Flows-Finite Difference, Element and Volume Techniques, ASME, New York, 1986, pp. 129-148. |

[8] | and , ’Finite element methods for the compressible Euler and Navier-Stokes equations, application to aerospace engineering’, First World Congr. on Computational Mechanics, Abstracts, Vol. 1, 1986. |

[9] | ’Characteristic Galerkin methods for hyperbolic problems’, in Proc. 5th GAMM Conf. on Numerical Methods in Fluid Mechanics, Rome, 1983. |

[10] | and , ’Approximation’ of nonlinear problems by least-squares finite elements, in (ed.), Nonlinear Analysis and Application, Marcel Dekker, New York, 1987. |

[11] | Carey, Int. j. numer. methods eng. 26 pp 81– (1988) |

[12] | Jiang, Int. j. numer. methods fluids 8 pp 933– (1988) |

[13] | and , ’Least-squares finite elements for convective transport problems’, Proc. Ninth SPE Symp. on Reservoir Simulation, Society of Petroleum Engineers (SPE), Inc., 1987, pp. 253-257. |

[14] | and , ’Least-squares finite elements for convective transport problems’, Advances in Water Resources, submitted. |

[15] | and , ’Least-squares finite elements for compressible Euler equations’, in , (eds), Numerical Methods in Laminar and Turbulent Flow, Vol. 5, Pineridge Press, Swansea, 1987, pp. 1460-1464. |

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.