A general methodology to simulate acoustic propagation in ducts with extended-reacting liners in the time domain is presented, including a generic perforated sheet on the air-material interface. The Linearized Euler Equations (LEE) with a mean flow profile are solved in the duct and the linearized equations on an equivalent fluid are solved in the liner material. The auxiliary differential equation method (ADE) is used to prevent the computation of convolution integrals, and leads to a formulation compatible with high-order numerical schemes. The methodology is illustrated for the case of liners consisting of rigid-frame porous materials and a prototype of locally-resonant acoustic metamaterials. A one-dimensional (1D) test case is first used to validate the algorithm and assess the numerical error. The numerical order of the algorithm is the expected one, independently of the interface, as long as the number of poles retained in the partial fraction expansions involved in the formulation is high enough. The algorithm is then applied to a realistic two-dimensional (2D) configuration in a duct with flow, and is used to illustrate the restrained validity of the locally-reacting approximation. Finally, the impact of the flow Mach number on the acoustic performance of porous and metamaterial extended-reacting liners is briefly assessed