The detonation combustion mode is very attractive for being used in new propulsion systems. The onset of detonation is a key issue for the process. In this work, a three-dimensional numerical simulation of the processes of reflection and focusing of a shock wave in a wedge was done. The following possible scenarios for the development of processes depending on the intensity of the incident shock wave were obtained: reflecting of shock wave without igniting, formation of the detonation wave in reflection and focusing, reflecting of the shock wave with lagging behind combustion zone and intermediate transient regimes with successive deflagration to detonation transition in lagging combustion zone. It was found that in the range of parameters of transient regimes when turbulence is taken into account the result of numerical simulation qualitatively differs from the result of the model without taking it into account. The model with turbulence and viscosity gives the correct picture in the range of parameters of transient regimes when using comparison with experimental data in contrast to the model disregarding turbulence. For other regimes, both models demonstrate good agreement with experiment.