### Abstract

The spatio-temporal structure of unstable detonations in a single space dimension is studied. A high resolution numerical method for computing unstable detonations is developed. This method combines the piecewise parabolic method (PPM) with conservative shock tracking and adaptive mesh refinement. A nonlinear asymptotic theory for the spatio-temporal growth of instabilities is also developed. This asymptotic theory involves a nonclassical Hopf bifurcation, because resonant acoustic scattering states with exponential growth in space cross the imaginary axis and become nonlinear eigenmodes in a complex free-boundary problem for a nonlinear hyperbolic equation. An interplay between the asymptotic theory and numerical simulation is used to elucidate the spatio-temporal mechanisms of nonlinear stability near the transition boundary; in particular, a quantitative-qualitative explanation is developed for the experimentally observed instabilities for supersonic blunt bodies advancing into appropriate reactive mixtures. The new numerical method is also used to predict regimes of multimode and chaotic pulsation instabilities. This numerical method is tested thoroughly on a classical unstable detonation problem.

Original language | English (US) |
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Pages (from-to) | 303-343 |

Number of pages | 41 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 51 |

Issue number | 2 |

State | Published - Apr 1991 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*51*(2), 303-343.