Time-Scale Decoupling for Nearly Periodic Advection-Diffusion Equations

An advection-diffusion problem is considered with an O(ɛ) perturbed T-periodic flow and O(ɛ) diffusivity, where ɛ > 0 denotes a small parameter. Such problems arise in tidally induced transport of constituents. An asymptotic approximation of the solution with O(ɛ) accuracy on a long O(l/ɛ)-time scale is constructed by using a rigorous averaging method. The averaged problem on the long time scale is completely decoupled from the O(1) time scale of the nearly periodic flow. As a consequence, numerical computations of the solution of the approximate problem can be based on a much larger time step than for the original problem. Thus a large factor is gained in numerical efficiency. In addition, piling up of discretization errors in the averaged problem is less severe than for numerical computations for the original problem. When this rigorous averaging procedure is used, two aspects of domain decomposition of the space-time cylinder play a role.