### Abstract

An efficient algorithm is described for calculating stationary one-dimensional transonic outflow solutions of the compressible Euler equations with gravity and heat source terms. The stationary equations are solved directly by exploiting their dynamical system form. Transonic expansions are the stable manifolds of saddle-point-type critical points, and can be obtained efficiently and accurately by adaptive integration outward from the critical points. The particular transonic solution and critical point that match the inflow boundary conditions are obtained by a two-by-two Newton iteration which allows the critical point to vary within the manifold of possible critical points. The proposed Newton Critical Point (NCP) method typically converges in a small number of Newton steps, and the adaptively calculated solution trajectories are highly accurate. A sample application area for this method is the calculation of transonic hydrodynamic escape flows from extrasolar planets and the early Earth. The method is also illustrated for an example flow problem that models accretion onto a black hole with a shock.

Original language | English |
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Pages (from-to) | 916-928 |

Number of pages | 13 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 223 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Jan 2009 |

Externally published | Yes |

### Keywords

- Compressible gas dynamics
- Critical points
- Dynamical systems
- Euler equations
- Newton method
- Transonic flow

## Cite this

*Journal of Computational and Applied Mathematics*,

*223*(2), 916-928. https://doi.org/10.1016/j.cam.2008.03.019