Ignition processes in the hydrogen-oxygen system were simulated by solving the corresponding conservation equations (i.e., conservation of mass, energy, momentum, and species mass) for one-dimensional geometries using a detailed reaction mechanism and a multispecies transport model. An additional source term in the energy conservation allowed the treatment of induced ignition, and a realistic model for the destruction of reactive species at the vessel surface was used to treat auto-ignitions in static reactors. Spatial discretization using finite differences and an adaptive grid point system led to a differential-algebraic equation system, which was solved numerically by extrapolation or by backward differencing codes. Comparisons with experimental works show that one common reaction mechanism is able to simulate shock-tube-induced ignitions (modeled by treating the reaction system as a homogeneous mixture heated up by the shock wave) as well as the three explosion limits of the hydrogen-oxygen system. Minimum ignition energies are calculated for various mixture compositions, pressures, radii of the external energy source, and ignition times, and it is shown that for long ignition times the "uniform pressure assumption" is a quite good approximation for computing minimum ignition energies.