Computer-aided analysis of the convergence to steady state of discrete approximations to the Euler equations.

*(English)*Zbl 0624.76078The behaviour of a centered finite volume scheme for the isoenergetic Euler equations in two space dimensions is studied by numerical differentiation and approximate eigensystem analysis. The entire semidiscrete approximation including boundary conditions is formulated as a large system of ordinary differential equations, which are linearized by numerically approximating the Fréchet derivative. An approximate eigensystem procedure that only needs the Fréchet derivative is used to extract the least damped eigenmodes. The overall method has been applied to the case of transonic flow past an airfoil and has revealed that the most persistent transient modes are highly structured and are associated with eigenvalues of small modulus. Furthermore, they appear to be centered around the shock region, the stagnation region and the trailing edge/wake region of the airfoil. The beneficial effect of local time-step scaling and artificial dissipation is also demonstrated by the method.

##### MSC:

76H05 | Transonic flows |

76N15 | Gas dynamics, general |

76M99 | Basic methods in fluid mechanics |

35Q30 | Navier-Stokes equations |

##### Keywords:

centered finite volume scheme; isoenergetic Euler equations; numerical differentiation; approximate eigensystem analysis; semidiscrete approximation; boundary conditions; Fréchet derivative; transonic flow past an airfoil
PDF
BibTeX
XML
Cite

\textit{L. E. Eriksson} and \textit{A. Rizzi}, J. Comput. Phys. 57, 90--128 (1985; Zbl 0624.76078)

Full Text:
DOI

##### References:

[1] | Lomax, H.; Pulliam, T.H.; Jespersen, D.C., Eigensystem analysis techniques for finite-difference equations. I. multilevel techniques, AIAA paper 81-1027, (1981) |

[2] | Rizzi, A.; Eriksson, L.E., () |

[3] | Eriksson, L.E., A study of mesh singularities and their effects on numerical errors, FFA tech. note TN 1984-10, (1984), Stockholm |

[4] | Denton, J.D., () |

[5] | Arnoldi, W.E., Quart. appl. math., 9, 17-29, (1951) |

[6] | Saad, Y., Linear algebra appl., 34, 269-295, (1980) |

[7] | Gary, J., Math. comp., 18, 1-18, (1964) |

[8] | Jameson, A.; Baker, T., AIAA paper 84-0093, (1984), New York |

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.