The optimized high-order Dispersion-Relation-Preserving finite difference scheme is extended so as to be able to compute multiple-scales aeroacoustics problems effectively and efficiently. It is proposed that a multi-size mesh is used in the computation. In implementing the time marching scheme, the computation domain is first partitioned into a number of subdomains. In each subdomain, a single size mesh is used. The mesh size of adjacent subdomain changes by a factor of two. The time step of adjacent subdomains also changes by the same ratio. This choice serves not only to maintain numerical stability but also to avoid unnecessary computations in regions with large size mesh. To pass information between subdomains, special optimized stencils are used at the subdomain interface region. Because rapid changes takes place at the mesh-size-change interfaces, they are likely sources of spurious numerical waves. To prevent the generation and spreading of spurious numerical waves, special artificial selective damping terms are developed for inclusion in the discretized scheme. As an illustration of the efficacy of the multi-size-mesh multi-time-step scheme, it is applied to the simulation of an automobile door cavity tone problem. The computed tone frequencies are found to agree well with experimental measurements.