The article is dedicated to the numerical investigation of gaseous pulsating detonation wave propagation using two approaches. In the first one the problem is solved in the laboratory frame and the detonation is initiated near the closed end of the channel. In the second approach the modeling is carried out in the shock-attached frame. For this purpose we proposed the numerical algorithm for the integration of shock evolution equation using a grid characteristic method. The algorithm is characterized by the second approximation order. The stable, weakly unstable, irregular and strongly unstable modes of detonation wave propagation are investigated using both approaches. The calculation of the stable mode demonstrates that the developed algorithm for the detonation wave modeling in the shock-attached frame has approximately the second approximation order indeed. The obtained results for the weakly unstable mode are in agreement with the linear theory. The qualitative and quantitative differences between two approaches are marked out. The most differences in results are noticed in the strongly unstable mode.