To accelerate the numerical simulation of compressible flows， a new acceleration methodology， named the Disturbance Region Update Method with Multigrid （DRUM-M）， is presented， which integrates the multigrid technique into the Disturbance Region Update Method （DRUM） proposed by the authors. The principles and algorithms of updating the Dynamic Computational Domains （DCDs） between the fine and the coarse grids are proposed. Improvements on certain operations， such as the initialization and the extension of the advective DCD， are made based on the characteristics of the multigrid technique. Besides， a new strategy of the coarse grid generation is established for structured grids， capable of eliminating the cell-number restriction of existing methods. Numerical test cases demonstrate that， firstly， DRUM-M can decrease the computational effort per iteration on both the fine and the coarse grids， as well as the total number of iterations； secondly， benefiting from employing DCDs on both the fine and the coarse grids， there are acceleration synergies between the multigrid technique and DRUM； thirdly， compared with the conventional multigrid technique at the same conditions， DRUM-M could achieve a time reduction of 57.9%.